‫ﻣﺴﺌﻠﻪ ﭼﺎﻧﻪﺯﻧﯽ ﺩﺭ ﺷﺒﮑﻪﻫﺎﯼ ﺍﺟﺘﻤﺎﻋﯽ‬
‫ﻣﻬﺮﺩﺍﺩ ﻣﻬﺪﻭﯼ‬
‫‪٨٧٣٠١٤٧٧‬‬
‫‪[email protected]‬‬
‫ﭼﮑﻴﺪﻩ‪ .‬ﻳﮑﯽ ﺍﺯ ﻣﺴﺎﻳﻞ ﻣﻬﻢ ﺩﺭ ﺗﺤﻠﻴﻞ ﻣﻌﺎﻣﻼﺕ ﺍﻗﺘﺼﺎﺩﯼ ﻭ ﺭﻭﺍﺑﻂ ﺳﻴﺎﺳﯽ‪ ،‬ﻣﺴﺌﻠﻪ ﭼﺎﻧﻪﺯﻧﯽ ﺍﺳﺖ ﮐﻪ ﺩﺭ ﺁﻥ‬
‫ﻃﺮﻓﻴﻦ ﻣﻌﺎﻣﻠﻪ ﺑﺮ ﺭﻭﯼ ﺍﺭﺯﺷﯽ ﺑﺮ ﺍﺳﺎﺱ ﺗﺎﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ﺧﻮﺩ ﺑﻪ ﭼﺎﻧﻪﺯﻧﯽ ﻣﯽﭘﺮﺩﺍﺯﻧﺪ ﺗﺎ ﺩﺭ ﺗﻘﺴﻴﻢ ﺍﺭﺯﺵ ﺑﻪ‬
‫ﺗﻮﺍﻓﻖ ﺑﺮﺳﻨﺪ‪ .‬ﺑﻌﺪ ﺍﺯ ﺭﺍﻩﺣﻞ ﻧﺶ ﺑﺮﺍﯼ ﺍﻳﻦ ﻣﺴﺌﻠﻪ ﺩﺭ ﺳﺎﻝ ‪ ،١٩٥٠‬ﺍﻳﻦ ﻣﺴﺌﻠﻪ ﺑﻴﻦ ﺍﻗﺘﺼﺎﺩﺩﺍﻧﺎﻥ ﻭ‬
‫ﺟﺎﻣﻌﻪﺷﻨﺎﺳﺎﻥ ﻣﻮﺭﺩ ﺗﻮﺟﻪ ﺑﻮﺩﻩ ﺍﺳﺖ ﻭ ﺭﺍﻩﺣﻞ ﻫﺎﻳﯽ ﺑﺮﺍﯼ ﺁﻥ ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﺩﺭ ﺳﺎﻝﻫﺎﯼ ﺍﺧﻴﺮ ﺑﺎ ﮔﺴﺘﺮﺵ‬
‫ﺷﺒﮑﻪﻫﺎﯼ ﺍﺟﺘﻤﺎﻋﯽ‪ ،‬ﺣﻞ ﻣﺴﺌﻠﻪ ﺩﺭ ﺷﺒﮑﻪﺍﯼ ﺍﺯ ﺭﻭﺍﺑﻂ ﺑﻴﻦ ﺍﻓﺮﺍﺩ ﻣﺎﻧﻨﺪ ﺭﻭﺍﺑﻂ ﺩﻭﺳﺘﯽ‪ ،‬ﺍﻗﺘﺼﺎﺩﯼ ﻭ ﺳﻴﺎﺳﯽ‬
‫ﻣﻮﺭﺩ ﺗﻮﺟﻪ ﻗﺮﺍﺭ ﮔﺮﻓﺘﻪ ﺍﺳﺖ ﻭ ﺍﺯ ﺩﻳﺪﮔﺎﻩ ﺗﺌﻮﺭﯼ ﺑﺎﺯﯼﻫﺎ ﻭ ﻋﻠﻢ ﮐﺎﻣﭙﻴﻮﺗﺮ ﺭﺍﻩﺣﻞﻫﺎﻳﯽ ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﺩﺭ ﺍﻳﻦ‬
‫ﻣﻘﺎﻟﻪ ﺑﻪ ﺑﺮﺭﺳﯽ ﻣﺴﺌﻠﻪ ﭼﺎﻧﻪﺯﻧﯽ ﺩﺭ ﺷﺒﮑﻪﻫﺎﯼ ﺍﺟﺘﻤﺎﻋﯽ ﻭ ﺗﺤﻠﻴﻞ ﺷﺮﺍﻳﻂ ﻭﺟﻮﺩ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻭ ﺗﺎﺛﻴﺮ‬
‫ﺳﺎﺧﺘﺎﺭ ﺷﺒﮑﻪ ﻭ ﻧﻮﻉ ﺗﺎﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ﺩﺭ ﺭﺍﻩﺣﻞ ﻧﻬﺎﻳﯽ ﻣﺴﺌﻠﻪ ﻣﯽﭘﺮﺩﺍﺯﻳﻢ‪.‬‬
‫‪ .١‬ﻣﻘﺪﻣﻪ‬
‫ﻣﺴﺌﻠﻪ ﭼﺎﻧﻪﺯﻧﯽ‪ ١‬ﻳﮑﯽ ﺍﺯ ﻣﺴﺎﻳﻞ ﻣﻬﻢ ﺩﺭ ﺟﺎﻣﻌﻪﺷﻨﺎﺳﯽ ﻭ ﺍﻗﺘﺼﺎﺩ ﺍﺳﺖ ﮐﻪ ﺩﺭ ﺩﻭ ﺳﺎﻝ ﺍﺧﻴﺮ ﺑﺎ ﺭﻭﻳﮑﺮﺩ ﻋﻠﻢ ﮐﺎﻣﭙﻴﻮﺗﺮ ﻭ‬
‫ﺗﺌﻮﺭﯼ ﺑﺎﺯﯼﻫﺎ ﻣﻮﺭﺩ ﺗﻮﺟﻪ ﺯﻳﺎﺩﯼ ﻗﺮﺍﺭ ﮔﺮﻓﺘﻪ ﺍﺳﺖ‪ .‬ﻳﮑﯽ ﺍﺯ ﮐﺎﺭﺑﺮﺩﻫﺎﯼ ﺍﻳﻦ ﻣﺴﺌﻠﻪ ﺩﺭ ﺑﺎﺯﯼﻫﺎﻳﯽ ﺍﺳﺖ ﮐﻪ ﺩﺍﺭﺍﯼ ﭼﻨﺪ ﻧﻘﻄﻪ‬
‫ﺗﻌﺎﺩﻝ ﻧﺶ ﻣﯽﺑﺎﺷﺪ‪ .‬ﺍﻳﻨﮑﻪ ﺩﺭ ﻧﻬﺎﻳﺖ ﮐﺪﺍﻡ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﺑﻪ ﻋﻨﻮﺍﻥ ﻧﺘﻴﺠﻪ ﻧﻬﺎﻳﯽ ﺍﻧﺘﺨﺎﺏ ﻣﯽﺷﻮﺩ ﺭﺍ ﻣﯽﺗﻮﺍﻥ ﺑﻪ ﺻﻮﺭﺕ ﻣﺴﺌﻠﻪ‬
‫ﭼﺎﻧﻪﺯﻧﯽ ﻣﺪﻝ ﻧﻤﻮﺩ‪ .‬ﻫﻤﭽﻨﻴﻦ ﺑﺴﻴﺎﺭﯼ ﺍﺯ ﻣﺴﺎﺋﻞ ﺑﺎﺯﺍﺭ ﺩﺭ ﺍﻗﺘﺼﺎﺩ ﺭﺍ ﻣﯽﺗﻮﺍﻥ ﺑﻪ ﺍﻳﻦ ﻣﺴﺌﻠﻪ ﺗﺒﺪﻳﻞ ﻧﻤﻮﺩ‪ .‬ﺩﻭ ﺑﺎﺯﻳﮑﻦ ‪ A‬ﻭ‬
‫‪ B‬ﺭﺍ ﺑﺎ ﺗﺎﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ‪ U A‬ﻭ ‪ U B‬ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻳﺪ ﮐﻪ ﺑﺮ ﺭﻭﯼ ﻳﮏ ﮐﺎﻻ ﻭ ﻳﺎ ﺗﻘﺴﻴﻢ ﻣﻘﺪﺍﺭﯼ ﺍﺭﺯﺵ ﻣﺎﻧﻨﺪ ﭘﻮﻝ ﭼﺎﻧﻪ‬
‫ﻣﯽﺯﻧﻨﺪ ﺗﺎ ﺑﻪ ﺗﻮﺍﻓﻖ ﺑﺮﺳﻨﺪ‪ .‬ﻫﻤﭽﻨﻴﻦ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺍﻧﺘﺨﺎﺏﻫﺎﯼ ﺩﻳﮕﺮ ‪ a A‬ﻭ ‪ a B‬ﺭﺍ ﻧﻴﺰ ﺩﺍﺭﻧﺪ ﮐﻪ ﺩﺭ ﺻﻮﺭﺕ ﻋﺪﻡ ﺗﻮﺍﻓﻖ ﺩﺭﻳﺎﻓﺖ‬
‫ﻣﯽﮐﻨﻨﺪ‪ .‬ﺑﺎﺯﻳﮑﻨﺎﻥ ﺑﺮ ﺭﻭﯼ ﺍﺭﺯﺵ ‪ c‬ﭼﺎﻧﻪ ﻣﯽﺯﻧﻨﺪ ﺗﺎ ﺁﻥ ﺭﺍ ﺑﻴﻦ ﺧﻮﺩ ﺗﻘﺴﻴﻢ ﮐﻨﻨﺪ‪ .‬ﺩﺭ ﻳﮏ ﻣﻌﺎﻣﻠﻪ ‪ c‬ﺭﺍ ﻣﯽﺗﻮﺍﻥ ﺍﺧﺘﻼﻑ‬
‫ﺑﻴﻦ ﻗﻴﻤﺖ ﻣﻮﺭﺩ ﻧﻈﺮ ﻓﺮﻭﺷﻨﺪﻩ ﻭ ﺍﺭﺯﺵ ﻣﻮﺭﺩ ﻧﻈﺮ ﺧﺮﻳﺪﺍﺭ ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺖ‪ .‬ﺍﮔﺮ ﭘﻴﺸﻨﻬﺎﺩ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺭﺍ ﺑﻪ ﺗﺮﺗﻴﺐ ﺑﺮﺍﺑﺮ ﺑﺎ ‪ x‬ﻭ‬
‫‪ y‬ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻳﻢ‪ ،‬ﻫﺮ ﺗﻘﺴﻴﻤﯽ ﮐﻪ ﺑﻪ ﺑﺎﺯﻳﮑﻦ ﺍﻭﻝ ﮐﻤﺘﺮ ﺍﺯ ) ‪ U A (a A‬ﻭ ﺑﻪ ﺑﺎﺯﻳﮑﻦ ﺩﻭﻡ ﮐﻤﺘﺮ ﺍﺯ ) ‪ U B (a B‬ﺍﺧﺘﺼﺎﺹ‬
‫ﻧﺪﻫﺪ ﻳﮏ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﺍﺳﺖ‪ .‬ﻫﻤﺎﻧﻄﻮﺭ ﮐﻪ ﻣﺸﺨﺺ ﺍﺳﺖ ﻓﻀﺎﯼ ﺣﺎﻟﺖ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﺑﯽﻧﻬﺎﻳﺖ ﻣﯽﺑﺎﺷﺪ‬
‫) ‪ .( x > U A (a A ), y > U B (a B ) , x + y = c‬ﺑﺮﺍﯼ ﺩﺍﺷﺘﻦ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻳﮑﺘﺎ‪ ،‬ﻧﺶ ﺩﺭ ﺳﺎﻝ ‪ ١٩٥٠‬ﺩﺭ ﻣﻘﺎﻟﻪ ]‪[١‬‬
‫ﺭﺍﻩﺣﻠﯽ ﺑﺮﺍﯼ ﺍﻳﻦ ﻣﺴﺌﻠﻪ ﺍﺭﺍﺋﻪ ﻧﻤﻮﺩ ﮐﻪ ﺑﻪ ﺭﺍﻩﺣﻞ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ‪ ٢‬ﻣﻌﺮﻭﻑ ﺍﺳﺖ ﻭ ﺩﺍﺭﺍﯼ ﺳﻪ ﻭﻳﮋﮔﯽ ‪ (١‬ﺛﺎﺑﺖ ﺩﺭ ﺑﺮﺍﺑﺮ‬
‫ﺗﻐﻴﻴﺮﺍﺕ ﺧﻄﯽ‪ (٢ ،‬ﺑﻬﻴﻨﻪ ‪ IIA٣(٣ ،Pareto‬ﻭ ‪ ( ٤‬ﻣﺘﻘﺎﺭﻥ ﺍﺳﺖ ]‪. [١‬‬
‫ﺩﺭ ﺭﺍﻩﺣﻞ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ‪ ،‬ﺭﺍﻩ ﺣﻞ ﻧﻬﺎﻳﯽ ﺗﻘﺴﻴﻢ ﺍﺭﺯﺵ ﺍﺿﺎﻓﯽ ﺑﻪ ﺻﻮﺭﺕ ﻋﺎﺩﻻﻧﻪ ﺑﻴﻦ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺍﺳﺖ ﮐﻪ ﻳﮏ ﺗﻌﺎﺩﻝ‬
‫ﭼﺎﻧﻪﺯﻧﯽ ﭘﺎﻳﺪﺍﺭ ﺍﻳﺠﺎﺩ ﻣﯽﮐﻨﺪ‪ .‬ﺩﺭ ﻭﺍﻗﻊ ﺭﺍﻩﺣﻞ ﻧﺶ ﺗﻘﺴﻴﻢ ﺍﺭﺯﺵ ﺭﺍ ﺑﻪ ﮔﻮﻧﻪﺍﯼ ﺍﻧﺠﺎﻡ ﻣﯽﺩﻫﺪ ﮐﻪ ﺿﺮﺏ ﺍﺧﺘﻼﻑ ﺳﻮﺩﻣﻨﺪﯼ‬
‫ﻣﻌﺎﻣﻠﻪ ﺭﺍ ﺑﺮﺍﯼ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﺑﻴﺸﻴﻨﻪ ﻣﯽﮐﻨﺪ‪ .‬ﺍﺧﺘﻼﻑ ﺳﻮﺩﻣﻨﺪﯼ ﺑﺮﺍﯼ ﻋﺎﻣﻞ ‪ A‬ﺍﺧﺘﻼﻑ ﺑﻴﻦ ﺳﻮﺩﯼ ﺍﺳﺖ ﮐﻪ ﺩﺭ ﺻﻮﺭﺕ‬
‫ﺗﻮﺍﻓﻖ ﺩﺭﻳﺎﻓﺖ ﻣﯽﮐﻨﺪ ﺑﺎ ﺳﻮﺩﯼ ﺍﺳﺖ ﮐﻪ ﺩﺭ ﺻﻮﺭﺕ ﻋﺪﻡ ﺗﻮﺍﻓﻖ ﺩﺭﻳﺎﻓﺖ ﻣﯽﮐﻨﺪ ﮐﻪ ﺑﺮﺍﺑﺮ ﺑﺎ ) ‪ U A ( x) - U A (a A‬ﮐﻪ ‪x‬‬
‫ﺳﻬﻢ ‪ A‬ﺍﺯ ﺗﻮﺍﻓﻖ ﺍﺳﺖ‪ .‬ﺑﻪ ﻫﻤﻴﻦ ﺗﺮﺗﻴﺐ ﺑﺮﺍﯼ ‪ B‬ﺑﺮﺍﺑﺮ ﺑﺎ ) ‪ U B (c - x) - U B (a B‬ﻣﯽﺑﺎﺷﺪ‪ .‬ﻫﺪﻑ ﺭﺍﻩﺣﻞ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ‬
‫ﺑﻴﺸﻴﻨﻪ ﮐﺮﺩﻥ )) ‪ (U A ( x) - U A (a A ))(U B (c - x) - U B (a B‬ﺍﺳﺖ‪ .‬ﺍﮔﺮ ﺗﺎﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺭﺍ ﺗﺎﺑﻊ ﺛﺎﺑﺖ ﺩﺭ ﻧﻈﺮ‬
‫ﺷﮑﻞ ‪ .١‬ﻳﮏ ﺷﺒﮑﻪ ﺍﺟﺘﻤﺎﻋﯽ ﺑﺎ ‪ ٥‬ﻧﻔﺮ ﮐﻪ ﺑﺎﺯﻳﮑﻦ ‪ v‬ﻗﻮﻳﺘﺮﻳﻦ ﻣﻮﻗﻌﻴﺖ ﺭﺍ ﺩﺍﺭﺩ‬
‫ﺑﮕﻴﺮﻳﻢ ) ‪ ( U ( x) = x‬ﺩﺭ ﺍﻳﻦ ﺻﻮﺭﺕ ﺭﺍﻩﺣﻞ ﻧﺶ ﻣﻘﺎﺩﻳﺮ ‪ a A + s / 2‬ﻭ ‪ a B + s / 2‬ﺑﻪ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺍﺧﺘﺼﺎﺹ ﻣﯽﺩﻫﺪ ﮐﻪ‬
‫‪ . s = c - a A - a B‬ﺭﺍﻩﺣﻞ ﺩﻳﮕﺮﯼ ﮐﻪ ﺑﺮﺍﯼ ﻣﺴﺌﻠﻪ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﺍﺳﺖ ﺭﺍﻩﺣﻞ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺴﺒﯽ ﺍﺳﺖ ﮐﻪ‬
‫ﺳﻌﯽ‬
‫ﺩﺭ‬
‫ﮐ ﻤﻴﻨ ﻪ‬
‫ﮐﺮﺩﻥ‬
‫ﺍﺧﺘﻼﻑ‬
‫ﺳﻮﺩﻣﻨﺪﯼ‬
‫ﺑﺎﺯﻳﮑﻨﺎﻥ‬
‫ﺍﺳﺖ‬
‫ﻳﻌﻨﯽ‬
‫})) ‪.[٢] min{(U A ( x) - U A (a A )), (U B (c - x) - U B (a B‬‬
‫ﻣﺴﺌﻠﻪﺍﯼ ﮐﻪ ﻣﺎ ﺩﺭ ﺍﻳﻦ ﻣﻘﺎﻟﻪ ﺑﻪ ﺁﻥ ﻣﯽﭘﺮﺩﺍﺯﻳﻢ ﺣﺎﻟﺖ ﺗﻌﻤﻴﻢ ﻳﺎﻓﺘﻪ ﻣﺴﺌﻠﻪ ﭼﺎﻧﻪﺯﻧﯽ ﺩﻭﻃﺮﻓﻪ ﺍﺳﺖ‪ .‬ﺩﺭ ﺍﻳﻦ ﺣﺎﻟﺖ ﻳﮏ‬
‫ﺳﺮﯼ ﺑﺎﺯﻳﮑﻦ ﺩﺍﺭﻳﻢ ﮐﻪ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﻣﯽﺗﻮﺍﻧﺪ ﺑﺎ ﻋﺪﻩﺍﯼ ﺍﺯ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺑﺮ ﺭﻭﯼ ﻳﮏ ﺳﺮﯼ ﮐﺎﻻﻫﺎ ﭼﺎﻧﻪﺯﻧﯽ ﮐﻨﺪ‪ .‬ﺩﺭ ﺍﻳﻦ ﺣﺎﻟﺖ ﻳﮏ‬
‫ﮔﺮﺍﻑ ﺍﺯ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺧﻮﺍﻫﻴﻢ ﺩﺍﺷﺖ ﮐﻪ ﺩﺭ ﺁﻥ ﺑﻴﻦ ﻫﺮ ﺩﻭ ﺑﺎﺯﻳﮑﻨﯽ ﮐﻪ ﺍﻣﮑﺎﻥ ﭼﺎﻧﻪﺯﻧﯽ ﺑﻴﻦ ﺁﻧﻬﺎ ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﻳﺎﻟﯽ ﺑﺎ ﻭﺯﻥ ﺍﺭﺯﺵ‬
‫ﮐﺎﻻﯼ ﭼﺎﻧﻪﺯﻧﯽ ﻗﺮﺍﺭ ﻣﯽﺩﻫﻴﻢ‪ .‬ﺩﻭ ﻭﻳﮋﮔﯽ ﻧﻘﺶ ﻣﻬﻢ ﺩﺭ ﺧﺮﻭﺟﯽ ﺍﻳﻦ ﺑﺎﺯﯼ ﺩﺍﺭﺩ‪ :‬ﺳﺎﺧﺘﺎﺭ ﺷﺒﮑﻪ ﻭ ﻧﻮﻉ ﺗﺎﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ‬
‫ﺑﺎﺯﻳﮑﻨﺎﻥ‪ .‬ﺗﻔﺎﻭﺗﯽ ﮐﻪ ﺍﻳﻦ ﻣﺴﺌﻠﻪ ﺑﺎ ﻣﺴﺌﻠﻪ ﭼﺎﻧﻪﺯﻧﯽ ﺩﻭ ﻃﺮﻓﻪ ﺩﺍﺭﺩ ﺍﻳﻦ ﺍﺳﺖ ﮐﻪ ﺳﺎﺧﺘﺎﺭ ﺷﺒﮑﻪ ﻧﻘﺶ ﺗﻌﻴﻴﻦ ﮐﻨﻨﺪﻩﺍﯼ ﺩﺭ‬
‫ﺧﺮﻭﺟﯽ ﻧﻬﺎﻳﯽ ﺑﺎﺯﯼ ﺧﻮﺍﻫﺪ ﺩﺍﺷﺖ‪ .‬ﺑﻪ ﻋﻨﻮﺍﻥ ﻣﺜﺎﻝ ﮔﺮﺍﻑ ﺷﮑﻞ ‪ ١‬ﺭﺍ ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻳﺪ ﮐﻪ ﻳﮏ ﺷﺒﮑﻪ ﺑﻴﻦ ‪ ٥‬ﻧﻔﺮ ﺑﺎ ﺭﻭﺍﺑﻂ ﺑﻴﻦ‬
‫ﺁﻧﻬﺎ ﺭﺍ ﻧﺸﺎﻥ ﻣﯽﺩﻫﺪ‪ .‬ﻓﺮﺽ ﮐﻨﻴﻢ ﮐﻪ ﻭﺯﻥ ﻫﺮ ﻳﺎﻝ )ﺍﺭﺯﺵ ﻫﺮ ﭼﺎﻧﻪﺯﻧﯽ( ﺑﺮﺍﺑﺮ ﺑﺎ ‪ ١‬ﺍﺳﺖ ﻭ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﺣﺪﺍﮐﺜﺮ ﺑﺎ ﻳﮏ ﺑﺎﺯﻳﮑﻦ‬
‫ﻣﯽﺗﻮﺍﻧﺪ ﺑﻪ ﺗﻮﺍﻓﻖ ﺑﺮﺳﺪ ﻭ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﺍﮔﺮ ﺑﻪ ﺗﻮﺍﻓﻘﯽ ﺩﺳﺖ ﻧﻴﺎﺑﺪ ﺳﻮﺩ ﺻﻔﺮ ﺩﺭﻳﺎﻓﺖ ﺧﻮﺍﻫﺪ ﮐﺮﺩ‪ .‬ﺍﺯ ﮔﺮﺍﻑ ﻣﺸﺨﺺ ﺍﺳﺖ ﮐﻪ‬
‫ﺩﺭ ﻧﻬﺎﻳﺖ ﺣﺪﺍﻗﻞ ﻳﮑﯽ ﺍﺯ ﺑﺎﺯﻳﮑﻨﺎﻥ ‪ u‬ﻭ ‪ w‬ﻣﻘﺪﺍﺭ ‪ ٠‬ﺩﺭﻳﺎﻓﺖ ﻣﯽﮐﻨﺪ‪ .‬ﻧﻮﺩ ‪ v‬ﺍﺯ ﺍﻳﻦ ﻗﺪﺭﺕ ﺧﻮﺩ ﺑﺮ ﺑﺎﺯﻳﮑﻨﺎﻥ ‪ u‬ﻭ ‪w‬‬
‫ﻣﯽﺗﻮﺍﻧﺪ ﺑﻪ ﻧﻔﻊ ﺧﻮﺩ ﺍﺳﺘﻔﺎﺩﻩ ﮐﻨﺪ ﻭ ﺗﻔﺮﻳﺒﺎ ﮐﻞ ﺍﺭﺯﺵ ﻳﮑﯽ ﺍﺯ ﻳﺎﻝﻫﺎﯼ )‪ (u, v‬ﻭ ﻳﺎ )‪ (v, w‬ﺭﺍ ﺩﺭﻳﺎﻓﺖ ﮐﻨﺪ‪ .‬ﺍﺯ ﻃﺮﻓﯽ ﻧﻮﺩ‬
‫‪ x‬ﻣﯽﺗﻮﺍﻧﺪ ﺩﺭﮎ ﮐﻨﺪ ﮐﻪ ﺍﺭﺯﺷﯽ ﺑﺮﺍﯼ ﻧﻮﺩ ‪ v‬ﻧﺪﺍﺭﺩ ﻭ ﻫﻤﻪ ﺍﻧﺮﮊﯼ ﺧﻮﺩ ﺭﺍ ﺑﺮﺍﯼ ﭼﺎﻧﻪﺯﻧﯽ ﺑﺎ ‪ y‬ﺻﺮﻑ ﮐﻨﺪ ﻭ ﺍﺭﺯﺵ‬
‫‪1‬‬
‫‪2‬‬
‫ﺩﺭ‬
‫ﺭﺍﻩﺣﻞ ﻧﺶ ﺑﻪ ﺩﺳﺖ ﺁﻭﺭﺩ‪.‬‬
‫ﺍﺯ ﺩﻳﺪﮔﺎﻩ ﺟﺎﻣﻌﻪﺷﻨﺎﺳﯽ ﻭ ﺍﻗﺘﺼﺎﺩﯼ ﺍﮐﺜﺮ ﮐﺎﺭﻫﺎﯼ ﺍﻧﺠﺎﻡ ﺷﺪﻩ ﺩﺭ ﺑﺎﺯﯼ ﭼﺎﻧﻪﺯﻧﯽ ﺩﺭ ﺷﺒﮑﻪﻫﺎﯼ ﺍﺟﺘﻤﺎﻋﯽ ﺑﺮ ﺭﻭﯼ ﺗﺤﻠﻴﻞ‬
‫ﻧﺘﺎﻳﺞ ﺗﺠﺮﺑﯽ ﺑﺮ ﺭﻭﯼ ﺷﺒﮑﻪﻫﺎﯼ ﺧﺎﺹ ﺑﻮﺩﻩ ﺍﺳﺖ‪ .‬ﺩﺭ ﺍﻳﻦ ﻣﻘﺎﻟﻪ ﻣﺎ ﺑﻪ ﺑﺮﺭﺳﯽ ﮐﺎﺭﻫﺎﯼ ﺗﺌﻮﺭﯼ ﺍﺯ ﺩﻳﺪﮔﺎﻩ ﻋﻠﻢ ﮐﺎﻣﭙﻴﻮﺗﺮ‬
‫ﻣﯽﭘﺮﺩﺍﺯﻳﻢ‪ .‬ﺩﺭ ]‪ [٣‬ﺭﺍﻩﺣﻠﯽ ﺑﺮ ﺍﺳﺎﺱ ﻗﺪﺭﺕ ﭼﺎﻧﻪﺯﻧﯽ ﺍﻓﺮﺍﺩ ﺩﺭ ﺷﺒﮑﻪ ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﺩﺭ ]‪ [٣‬ﺑﺮ ﺍﺳﺎﺱ ﺳﺎﺧﺘﺎﺭ ﺷﺒﮑﻪ ﺑﻪ ﻫﺮ‬
‫ﺑﺎﺯﻳﮑﻦ ﻳﮏ ﻣﻘﺪﺍﺭ ﻗﺪﺭﺕ ﭼﺎﻧﻪﺯﻧﯽ ﺑﺮ ﺍﺳﺎﺱ ﺭﻭﺍﺑﻂ ﻭﯼ ﺑﺎ ﺳﺎﻳﺮ ﺑﺎﺯﻳﮑﻨﺎﻥ ﻣﺤﺎﺳﺒﻪ ﻣﯽﺷﻮﺩ ﻭ ﺳﭙﺲ ﺑﺮ ﺍﺳﺎﺱ ﺍﻳﻦ ﻗﺪﺭﺕ ﺳﻬﻢ‬
‫ﺑﺎﺯﻳﮑﻦ ﺍﺯ ﻫﺮ ﭼﺎﻧﻪﺯﻧﯽ ﻣﺤﺎﺳﺒﻪ ﻣﯽﺷﻮﺩ‪ .‬ﺩﺭ ]‪ [٤‬ﻣﺴﺌﻠﻪ ﺑﺮﺍﯼ ﮔﺮﺍﻑﻫﺎﯼ ﺩﻭﺑﺨﺸﯽ ﺣﻞ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﻣﺤﺪﻭﺩﻳﺘﯽ ﮐﻪ ﺩﺭ ]‪ [٤‬ﺩﺭ‬
‫ﻧﻈﺮ ﮔﺮﻓﺘﻪ ﺷﺪﻩ ﺍﻳﻦ ﺍﺳﺖ ﮐﻪ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﺣﺪﺍﮐﺜﺮ ﺑﺎ ﻳﮏ ﺑﺎﺯﻳﮑﻦ ﻣﯽﺗﻮﺍﻧﺪ ﺑﺎ ﺗﻮﺍﻓﻖ ﺑﺮﺳﺪ‪ .‬ﺩﺭ ﻫﺮ ﺩﻭ ﻣﻘﺎﻟﻪ ]‪ [٣‬ﻭ ]‪ [٤‬ﺗﻮﺍﺑﻊ‬
‫ﺳﻮﺩﻣﻨﺪﯼ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺧﻄﯽ ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺘﻪ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﺩﺭ ]‪ [٥‬ﻣﺴﺌﻠﻪ ﺩﺭ ﺣﺎﻟﺖ ﻋﻤﻮﻣﯽ ﺷﺒﮑﻪﻫﺎ ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺘﻪ ﺷﺪﻩ ﻭ‬
‫ﺭﺍﻩﺣﻞﻫﺎﻳﯽ ﺑﺮﺍﯼ ﻣﺤﺎﺳﺒﻪ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﺍﺳﺖ‪.‬ﺩﺭ ]‪ [٥‬ﻣﺴﺌﻠﻪ ﺩﺭ ﺣﺎﻟﺖ ﮐﻠﯽ ﺣﻞ ﺷﺪﻩ ﻭ ﺭﺍﻩﺣﻞﻫﺎﻳﯽ ﺑﺮﺍﯼ‬
‫ﮔﺮﺍﻑﻫﺎﯼ ﺧﺎﺹ ﻭ ﺗﻮﺍﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ﻣﺘﻔﺎﻭﺕ ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﺩﺭ ﺍﺩﺍﻣﻪ ﺑﻌﺪ ﺍﺯ ﺍﺭﺍﺋﻪ ﺑﺮﺧﯽ ﻣﻘﺪﻣﺎﺕ ﺑﻪ ﺑﺮﺭﺳﯽ ﮐﺎﺭﻫﺎﯼ ﺍﻧﺠﺎﻡ‬
‫ﺷﺪﻩ ﻣﯽﭘﺮﺩﺍﺯﻳﻢ‪.‬‬
‫‪ .٢‬ﺗﻌﺎﺭﻳﻒ ﺍﻭﻟﻴﻪ‬
‫ﻭﺭﻭﺩﯼ ﺑﺎﺯﯼ ﭼﺎﻧﻪﺯﻧﯽ ﺩﺭ ﺷﺒﮑﻪ ﻳﮏ ﮔﺮﺍﻑ ) ‪ G (V , E‬ﺍﺳﺖ ﮐﻪ ‪ | V |= n‬ﻭ ‪ . | E |= m‬ﻳﺎﻝﻫﺎ ﻭﺯﻥ ﺩﺍﺭ ﺑﻮﺩﻩ ﻭ ﻭﺯﻥ ﻫﺮ‬
‫ﻳﺎﻝ ﺑﺮﺍﺑﺮ ﺑﺎ )‪ c(e‬ﻣﯽﺑﺎﺷﺪ‪ .‬ﺩﺭ ﺍﻳﻦ ﮔﺮﺍﻑ ﺭﺍﺱﻫﺎ ﻣﻌﺎﺩﻝ ﺑﺎﺯﻳﮑﻨﺎﻥ ﻭ ﻫﺮ ﻳﺎﻝ ﺑﺎ ﻭﺯﻧﺶ ﺍﺭﺯﺷﯽ ﺍﺳﺖ ﮐﻪ ﺩﻭ ﺑﺎﺯﻳﮑﻦ ﻣﺠﺎﻭﺭ ﺁﻥ‬
‫ﻳﺎﻝ ﻣﯽﺗﻮﺍﻧﺪ ﺑﺮ ﺭﻭﯼ ﺁﻥ ﺑﻪ ﭼﺎﻧﻪﺯﻧﯽ ﺑﭙﺮﺩﺍﺯﻧﺪ‪ .‬ﺑﺮﺍﯼ ﻫﺮ ﺑﺎﺯﯼ‪ ،‬ﺣﺎﻟﺖ ﻣﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﺗﻮﺍﻓﻘﺎﺕ ﺍﻧﺠﺎﻡ ﺷﺪﻩ ﺑﻪ ﺍﺯﺍﯼ ﻫﻤﻪ ﻳﺎﻟﻬﺎ ﺭﺍ‬
‫ﺑﺎ ‪ s = ( s1 , s2 ,, sm ) ÎÂm‬ﻧﺸﺎﻥ ﻣﯽﺩﻫﻴﻢ‪ .‬ﺍﮔﺮ ﺑﺮﺍﯼ ﻫﺮ ﻳﺎﻝ ) ‪ ei = (u , v‬ﻳﮏ ﺟﻬﺖ ﻓﺮﺿﯽ ﺍﺯ ‪ u‬ﺑﻪ ‪ v‬ﺩﺭ ﻧﻈﺮ‬
‫ﺑﮕﻴﺮﻳﻢ‪ si ،‬ﺳﻬﻢ ﺑﺎﺯﻳﮑﻦ ‪ u‬ﺭﺍ ﺍﺯ ﺍﺭﺯﺵ ) ‪ c(ei‬ﻧﺸﺎﻥ ﻣﯽﺩﻫﺪ ﻭ ﺑﺮﺍﺑﺮ ﺑﺎ ) ‪ x(u , ei‬ﺍﺳﺖ‪ .‬ﺑﺪﻳﻬﯽ ﺍﺳﺖ ﮐﻪ‬
‫) ‪ . x(v, ei ) = c(ei ) - x(u , ei‬ﻫﺮ ﭼﺎﻧﻪﺯﻧﯽ ﺑﻴﻦ ﺑﺎﺯﻳﮑﻨﺎﻥ ﻣﯽﺗﻮﺍﻧﺪ ﺑﺮ ﺍﺳﺎﺱ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﻭ ﻳﺎ ﻧﺴﺒﯽ ﺻﻮﺭﺕ ﮔﻴﺮﺩ ﮐﻪ‬
‫ﺗﻮﺍﻓﻖ ﺩﺭ ﻧﻬﺎﻳﺖ ﺑﺮ ﺍﺳﺎﺱ ﻧﻮﻉ ﺭﺍﻩﺣﻞ ﭼﺎﻧﻪﺯﻧﯽ ﻣﺘﻔﺎﻭﺕ ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪.‬‬
‫ﺣﺎﻝ ﺑﻪ ﺗﻌﺎﺭﻳﻒ ﺯﻳﺮ ﺩﺭ ﺑﺎﺯﯼ ﭼﺎﻧﻪﺯﻧﯽ ﻣﯽﭘﺮﺩﺍﺯﻳﻢ‪.‬‬
‫ﺗﻌﺮﻳﻒ ‪ .١‬ﻓﺮﺽ ﮐﻨﻴﻢ ﮐﻪ ‪ s Î Âm‬ﺣﺎﻟﺘﯽ ﺍﺯ ﭼﺎﻧﻪﺯﻧﯽ ﺩﺭ ﮔﺮﺍﻑ ‪ G‬ﺑﺎﺷﺪ‪ .‬ﺑﺮﺍﯼ ﻫﺮ ﺭﺍﺱ ‪ g s (u ) ، u‬ﺭﺍ ﺑﺮﺍﺑﺮ ﺑﺎ ﮐﻞ ﺳﻮﺩﯼ‬
‫ﮐﻪ ﺭﺍﺱ ‪ u‬ﺍﺯ ﺗﻮﺍﻓﻖ ﺑﺎ ﻫﻤﺴﺎﻳﻪ ﻫﺎﻳﺶ ﺑﻪ ﺩﺳﺖ ﺁﻭﺭﺩﻩ ﺍﺳﺖ ﺗﻌﺮﻳﻒ ﻣﯽﮐﻨﻴﻢ‪ .‬ﻫﻤﭽﻨﻴﻦ )‪ xs (u, e‬ﺑﺮﺍﺑﺮ ﺑﺎ ﺳﻮﺩﯼ ﺍﺳﺖ ﮐﻪ‬
‫‪ u‬ﺍﺯ ﺗﻮﺍﻓﻖ ﺭﻭﯼ ﻳﺎﻝ ‪ e‬ﺑﻪ ﺩﺳﺖ ﺁﻭﺭﺩﻩ ﺍﺳﺖ‪ a s (u, e) = g s (u ) - xs (u, e) .‬ﺭﺍ ﺑﺮﺍﺑﺮ ﺑﺎ ﺳﻮﺩ ﺑﺎﺯﻳﮑﻦ ‪ u‬ﺑﻪ ﺟﺰ ﻳﺎﻝ ‪e‬‬
‫ﺗﻌﺮﻳﻒ ﻣﯽﮐﻨﻴﻢ‪.‬‬
‫ﺩﺭ ﻭﺍﻗﻊ )‪ a s (u , e‬ﻭ )‪ a s (v, e‬ﺑﻪ ﻋﻨﻮﺍﻥ ﮔﺰﻳﻨﻪﻫﺎﯼ ﺩﻳﮕﺮ ﺑﺎﺯﻳﮑﻨﺎﻥ ‪ u‬ﻭ ‪ v‬ﺩﺭ ﺣﺎﻟﺘﯽ ﺍﺳﺖ ﮐﻪ ﺭﻭﯼ ﻳﺎﻝ )‪ e = (u, v‬ﺑﻪ‬
‫ﭼﺎﻧﻪﺯﻧﯽ ﻣﯽﭘﺮﺩﺍﺯﻧﺪ‪ .‬ﺑﻪ ﻋﺒﺎﺭﺕ ﺩﻳﮕﺮ )‪ a s (u , e‬ﻭ )‪ a s (v, e‬ﺳﻮﺩ ﺑﺎﺯﻳﮑﻨﺎﻥ ‪ u‬ﻭ ‪ v‬ﺑﻪ ﺍﺯﺍﯼ ﺣﺎﻟﺘﯽ ﺍﺳﺖ ﮐﻪ ﺩﺭ ﻳﺎﻝ ‪ e‬ﺑﻪ‬
‫ﺗﻮﺍﻓﻖ ﻧﺮﺳﻨﺪ‪.‬‬
‫ﺗﻌﺮﻳﻒ ‪ .٢‬ﻓﺮﺽ ﮐﻨﻴﻢ ﮐﻪ ‪ s Î Âm‬ﺣﺎﻟﺘﯽ ﺍﺯ ﭼﺎﻧﻪﺯﻧﯽ ﺩﺭ ﮔﺮﺍﻑ ‪ G‬ﺑﺎﺷﺪ‪ x .‬ﺭﺍ ﺳﻮﺩﯼ ﺗﻌﺮﻳﻒ ﻣﯽﮐﻨﻴﻢ ﮐﻪ ﺑﺎﺯﻳﮑﻦ ‪ u‬ﺍﺯ‬
‫ﺗﻮﺍﻓﻖ ﺭﻭﯼ ﻳﺎﻝ‬
‫)‪ e = (u , v‬ﺑﻪ ﺩﺳﺖ ﻣﯽﺁﻭﺭﺩ‪ .‬ﺍﺧﺘﻼﻑ ﺳﻮﺩ ﺑﺎﺯﻳﮑﻦ‬
‫))‪as (u ) =U u (a s (u, e) + x) -U u (a s (u , e‬‬
‫ﻭ‬
‫ﺍﺧﺘﻼﻑ‬
‫ﺳﻮﺩ‬
‫‪ u‬ﺍﺯ ﺍﻳﻦ ﺗﻮﺍﻓﻖ ﺑﺮﺍﺑﺮ ﺑﺎ‬
‫ﺑﺎﺯﻳﮑﻦ‬
‫‪v‬‬
‫ﺑﺮﺍﺑﺮ‬
‫ﺑﺎ‬
‫))‪ bs (v) = U v (a s (v, e) + c( x) - x) - U v (a s (v, e‬ﺍﺳﺖ‪ .‬ﺩﺭ ﺭﺍﻩ ﺣﻞ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﺑﺎﻳﺪ )‪ as (u ) as (v‬ﺑﻴﺸﻴﻨﻪ‬
‫ﺷﻮﺩ‪.‬‬
‫ﺗﻌﺮﻳﻒ ‪ .٣‬ﻓﺮﺽ ﮐﻨﻴﻢ ﺩﺭ ﺣﺎﻟﺖ ‪ s Î Âm‬ﺑﺎﺯﻳﮑﻦ ‪ u‬ﺗﻤﺎﻳﻞ ﺩﺍﺭﺩ ﺑﺮ ﺭﻭﯼ ﻳﺎﻝ )‪ e = (u , v‬ﺩﻭﺑﺎﺭﻩ ﺑﻪ ﭼﺎﻧﻪﺭﻧﯽ ﺑﭙﺮﺩﺍﺯﺩ ﺑﻪ‬
‫ﻧﺤﻮﯼ ﮐﻪ ﺗﻮﺍﻓﻖﻫﺎﯼ ﺍﻧﺠﺎﻡ ﺷﺪﻩ ﺑﺮ ﺭﻭﯼ ﺳﺎﻳﺮ ﻳﺎﻟﻬﺎ ﺛﺎﺑﺖ ﺑﻤﺎﻧﺪ‪ .‬ﺩﺭ ﻭﺍﻗﻊ ﺍﻳﻦ ﻣﻌﺎﺩﻝ ﺗﻐﻴﻴﺮ ﺍﺳﺘﺮﺍﺗﮋﯼ ﻳﮏ ﺑﺎﺯﻳﮑﻦ ﺩﺭ ﺑﺎﺯﯼﻫﺎﯼ‬
‫ﻣﻌﻤﻮﻟﯽ ﺍﺳﺖ‪ ys (u , e) .‬ﺭﺍ ﺑﺮﺍﺑﺮ ﺑﺎ ﮐﻞ ﺳﻮﺩ ﺑﺎﺯﻳﮑﻦ ﺑﻌﺪ ﺍﺯ ﺍﻧﺠﺎﻡ ﭼﺎﻧﻪﺯﻧﯽ ﺩﻭﺑﺎﺭﻩ ﺗﻌﺮﻳﻒ ﻣﯽﮐﻨﻴﻢ‪ .‬ﺩﺭ ﺍﻳﻦ ﺣﺎﻟﺖ ﺗﻐﻴﻴﺮ ﺳﻮﺩ‬
‫ﺑﺎﺯﻳﮑﻦ ﺑﺮﺍﺑﺮ ﺑﺎ | )‪ update( s, e) =| xs (u , e) - ys (u , e‬ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪.‬‬
‫ﺗﻌﺮﻳﻒ ‪ .٤‬ﺗﻮﺍﻓﻖ ﺑﺮ ﺭﻭﯼ ﻳﺎﻝ )‪ e = (u , v‬ﭘﺎﻳﺪﺍﺭ ﺍﺳﺖ ﺍﮔﺮ ﺗﻘﺴﻴﻢ ﺍﺭﺯﺵ ﻳﺎﻝ ﺑﺮﺍﯼ ﻫﺮ ﺩﻭ ﺑﺎﺯﻳﮑﻦ ﺭﺍﺿﯽ ﮐﻨﻨﺪﻩ ﺑﺎﺷﺪ‪ .‬ﺩﻭ‬
‫ﺑﺎﺯﻳﮑﻦ ﺯﻣﺎﻧﯽ ﺭﺍﺿﯽ ﻣﯽﺷﻮﻧﺪ ﮐﻪ ﺗﻘﺴﻴﻢ ﺳﻮﺩ ﺑﺮ ﺍﺳﺎﺱ ﺭﺍﻩﺣﻞ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﻭ ﻳﺎ ﻧﺴﺒﯽ ﺻﻮﺭﺕ ﮔﻴﺮﺩ‪ .‬ﺣﺎﻟﺖ ‪ s Î Âm‬ﺭﺍ‬
‫ﻳﮏ ﺣﺎﻟﺖ ﺗﻌﺎﺩﻝ ﻣﯽﻧﺎﻣﻴﻢ ﺍﮔﺮ ﻫﻤﻪ ﻳﺎﻝﻫﺎﯼ ﺁﻥ ﭘﺎﻳﺪﺍﺭ ﺑﺎﺷﻨﺪ ) ‪ .( update( s, e) = 0‬ﺑﺪﻳﻬﯽ ﺍﺳﺖ ﮐﻪ ﺍﮔﺮ ﺣﺎﻟﺘﯽ ﺗﻌﺎﺩﻝ‬
‫ﻧﺒﺎﺷﺪ‪ ،‬ﺑﻌﻀﯽ ﺍﺯ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺗﻤﺎﻳﻞ ﺩﺍﺭﻧﺪ ﺗﺎ ﺑﺮ ﺭﻭﯼ ﺑﺮﺧﯽ ﻳﺎﻟﻬﺎ ﺑﻪ ﭼﺎﻧﻪﺯﻧﯽ ﺑﭙﺮﺩﺍﺯﻧﺪ‪.‬‬
‫ﻟﻢ ‪ .١‬ﻓﺮﺽ ﮐﻨﻴﻢ ﺗﺎﺑﻊ ﺳﻮﻣﻨﺪﯼ ﻫﻤﻪ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺍﻓﺰﺍﻳﺸﯽ‪ ،‬ﻣﺤﺪﺏ‪ ،‬ﭘﻴﻮﺳﺘﻪ‪ ،‬ﻭ ﺩﻭ ﺑﺎﺭ ﻣﺸﺘﻖ ﭘﺬﻳﺮ ﺑﺎﺷﺪ‪ .‬ﻫﻤﭽﻨﻴﻦ ﻗﺮﺍﺭ‬
‫)‪a s ( x‬‬
‫= )‪ q s ( x‬ﻭ‬
‫) ‪bs ( x‬‬
‫= )‪ . rs ( x‬ﺩﺭ ﺍﻳﻦ ﺻﻮﺭﺕ ﺷﺮﻁ ﺗﻌﺎﺩﻝ ﻧﺶ ﺑﺎ )‪q s ( x) = rs ( x‬‬
‫ﻣﯽﺩﻫﻴﻢ‬
‫)‪b s ( x‬‬
‫)‪a ' s ( x‬‬
‫ﻣﻌﺎﺩﻝ ﺍﺳﺖ ﻭ ‪ x‬ﻳﮑﺘﺎﻳﯽ ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﮐﻪ ﺩﺭ ﺍﻳﻦ ﺷﺮﻁ ﺻﺪﻕ ﮐﻨﺪ‪.‬‬
‫'‬
‫‪ .٤‬ﭼﺎﻧﻪﺯﻧﯽ ﺑﺎ ﻣﺤﺪﻭﺩﻳﺖ ﺣﺪﺍﮐﺜﺮ ﻳﮏ ﺗﻮﺍﻓﻖ‬
‫ﺩﺭ ]‪ [٤‬ﻣﺴﺌﻠﻪ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﺩﺭ ﮔﺮﺍﻑﻫﺎﯼ ﺩﻭ ﺑﺨﺸﯽ ﺑﺮﺭﺳﯽ ﺷﺪﻩ ﺍﺳﺖ ﻭ ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﺷﺪﻩ ﺍﺳﺖ ﮐﻪ ﺩﺭ ﻫﺮ ﮔﺮﺍﻑ‬
‫ﺩﻭ ﺑﺨﺸﯽ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﻭﺟﻮﺩ ﺩﺍﺭﺩ‪ .‬ﺍﻟﺒﺘﻪ ﺍﻳﻦ ﺍﻣﺮ ﺑﺮﺍﯼ ﺣﺎﻟﺘﯽ ﮐﻪ ﻓﺮﺽ ﺷﺪﻩ ﺍﺳﺖ ﺗﺎﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺧﻄﯽ‬
‫ﺍﺳﺖ ﻭ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﺣﺪﺍﮐﺜﺮ ﻣﯽﺗﻮﺍﻧﺪ ﺑﺎ ﻳﮏ ﺑﺎﺯﻳﮑﻦ ﺑﻪ ﺗﻮﺍﻓﻖ ﺑﺮﺳﺪ ﺻﺎﺩﻕ ﺍﺳﺖ‪ .‬ﻫﻤﭽﻨﻴﻦ ﻓﺮﺽ ﺷﺪﻩ ﺍﺳﺖ ﮐﻪ ﻭﺯﻥ ﻫﺮ ﻳﺎﻝ‬
‫ﻳﮏ ﺍﺳﺖ‪ .‬ﺩﺭ ﺍﺩﺍﻣﻪ ﺍﺑﺘﺪﺍ ﺑﻪ ﺣﻞ ﻣﺴﺌﻠﻪ ﺩﺭ ﮔﺮﺍﻑﻫﺎﯼ ﺩﻭﺑﺨﺸﯽ ﺑﺎ ﺗﻄﺎﺑﻖ ﮐﺎﻣﻞ ﻳﮑﺘﺎ ﻣﯽﭘﺮﺩﺍﺯﻳﻢ ﻭ ﺳﭙﺲ ﻧﺘﺎﻳﺞ ﺭﺍ ﺑﻪ ﺣﺎﻟﺖ‬
‫ﻋﻤﻮﻣﯽ ﺩﺭ ﮔﺮﺍﻑﻫﺎﯼ ﺩﻭﺑﺨﺸﯽ ﺗﻌﻤﻴﻢ ﻣﯽﺩﻫﻴﻢ‪ .‬ﺩﺭ ﺣﺎﻟﺘﻴﮑﻪ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﻣﺠﺎﺯ ﺑﻪ ﺣﺪﺍﮐﺜﺮ ﻳﮏ ﺗﻮﺍﻓﻖ ﺑﺎﺷﺪ ﺧﺮﻭﺟﯽ ﺑﺎﺯﯼ‬
‫ﭼﺎﻧﻪﺯﻧﯽ ﻳﮏ ﺗﻄﺎﺑﻖ ﺩﺭ ﮔﺮﺍﻑ ﺩﻭﺑﺨﺸﯽ ﺑﺎﺯﯼ ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪ .‬ﺑﻪ ﺍﺯﺍﯼ ﻫﺮ ﻳﺎﻝ )‪ (u , w‬ﺩﺭ ﺗﻄﺎﺑﻖ‪ ،‬ﻧﻮﺩ ‪ u‬ﻣﻘﺪﺍﺭ ‪ g u‬ﻭ ﻧﻮﺩ ‪w‬‬
‫ﻣﻘﺪﺍﺭ ‪ g w‬ﺩﺭﻳﺎﻓﺖ ﻣﯽﮐﻨﺪ ﮐﻪ ‪ . g u + g w = 1‬ﺑﺮﺍﯼ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ‪ u‬ﮔﺰﻳﻨﻪ ﺩﻳﮕﺮ ﺁﻥ ﺩﺭ ﺑﺎﺯﯼ ﺑﻪ ﺍﺯﺍﯼ ﺗﻮﺍﻓﻖ )‪ (u , w‬ﺑﺮﺍﺑﺮ ﺑﺎ‬
‫‪ a u‬ﺍﺳﺖ ﮐﻪ ﺑﺮﺍﺑﺮ ﺑﺎ ﻣﻘﺪﺍﺭﯼ ﺍﺳﺖ ﮐﻪ ﻣﯽﺗﻮﺍﻧﺪ ﺍﺯ ﺗﻮﺍﻓﻖ ﺑﺎ ﺳﺎﻳﺮ ﻧﻮﺩﻫﺎﯼ ﻫﻤﺴﺎﻳﻪ ﺑﻪ ﺟﺰ ﻧﻮﺩ ‪ w‬ﺑﻪ ﺩﺳﺖ ﺁﻭﺭﺩ‪ .‬ﺍﮔﺮ‬
‫‪ a u £ g u‬ﺑﺎﺷﺪ‪ ،‬ﺗﻮﺍﻓﻖ ﭘﺎﻳﺪﺍﺭ ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪ .‬ﺧﺮﻭﺟﯽ ﺑﺎﺯﯼ ﭼﺎﻧﻪﺯﻧﯽ ﺩﺭ ﺻﻮﺭﺗﯽ ﭘﺎﻳﺪﺍﺭ ﺍﺳﺖ ﮐﻪ ﻫﻤﻪ ﺗﻮﺍﻓﻖﻫﺎ ﭘﺎﻳﺪﺍﺭ ﺑﺎﺷﺪ‪.‬‬
‫ﺧﺮﻭﺟﯽ ﭘﺎﻳﺪﺍﺭ ﺭﺍ ﻳﮏ ﺧﺮﻭﺟﯽ ﻣﺘﻮﺍﺯﻥ ﻣﯽﻧﺎﻣﻴﻢ ﺍﮔﺮ ﺗﻮﺍﻓﻖﻫﺎ ﺑﺮ ﺍﺳﺎﺱ ﺭﺍﻩﺣﻞ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﺻﻮﺭﺕ ﮔﻴﺮﺩ‪.‬‬
‫‪ .١ .٤‬ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﺩﺭ ﮔﺮﺍﻑ ﺩﻭﺑﺨﺸﯽ ﺑﺎ ﺗﻄﺎﺑﻖ ﮐﺎﻣﻞ ﻳﮑﺘﺎ‬
‫ﺩﺭ ﭼﻨﻴﻦ ﮔﺮﺍﻓﻬﺎﻳﯽ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﻭ ﻣﯽﺗﻮﺍﻧﺪ ﭼﻨﻴﻦ ﺗﻌﺎﺩﻟﯽ ﺭﺍ ﺑﻪ ﺻﻮﺭﺕ ﺗﮑﺮﺍﺭﯼ ﺩﺭ ﺯﻣﺎﻥ‬
‫ﭼﻨﺪﺟﻤﻠﻪﺍﯼ ﭘﻴﺪﺍ ﮐﺮﺩ‪ .‬ﺍﻳﺪﻩ ﺍﺻﻠﯽ ﺍﺛﺒﺎﺕ ﺑﻪ ﺍﻳﻦ ﺻﻮﺭﺕ ﺍﺳﺖ‪ .‬ﮔﺮﺍﻑ ﺩﻭ ﺑﺨﺸﯽ ) ‪ G (V , E‬ﺭﺍ ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻳﺪ ﮐﻪ ﻧﻮﺩﻫﺎﯼ‬
‫ﺑﺨﺶ ﺍﻭﻝ ﺭﺍ ﺑﺎ ‪ X‬ﻭ ﻧﻮﺩﻫﺎﯼ ﺑﺨﺶ ﺩﻭﻡ ﺭﺍ ﺑﺎ ‪ Y‬ﻧﻤﺎﻳﺶ ﻣﯽﺩﻫﻴﻢ‪ .‬ﻓﺮﺽ ﮐﻨﻴﻢ ‪ M‬ﺗﻄﺎﺑﻖ ﮐﺎﻣﻞ ﻳﮑﺘﺎ ﺩﺭ ﮔﺮﺍﻑ ﺑﺎﺷﺪ‪ .‬ﺍﺯ‬
‫ﮔﺮﺍﻑ ‪ G‬ﻳﮏ ﮔﺮﺍﻑ ﺟﻬﺖﺩﺍﺭ ﺍﻳﺠﺎﺩ ﻣﯽﮐﻨﻴﻢ‪ .‬ﺑﻪ ﺍﺯﺍﯼ ﻳﺎﻟﻬﺎﻳﯽ ﮐﻪ ﺩﺭ ‪ M‬ﻗﺮﺍﺭ ﺩﺍﺭﻧﺪ ﺟﻬﺖ ﻳﺎﻝ ﺭﺍ ﺍﺯ ‪ X‬ﺑﻪ ‪ Y‬ﻭ ﺑﺮﺍﯼ‬
‫ﺳﺎﻳﺮ ﻳﺎﻟﻬﺎ ﺍﺯ ‪ Y‬ﺑﻪ ‪ X‬ﻗﺮﺍﺭ ﻣﯽﺩﻫﻴﻢ‪ .‬ﭼﻮﻥ ﺗﻄﺎﺑﻖ ﮐﺎﻣﻞ ﻳﮑﺘﺎﺳﺖ ﮔﺮﺍﻑ ﺟﻬﺖﺩﺍﺭ ﺑﺪﻭﻥ ﺩﻭﺭ ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪ .‬ﺭﺍﺑﻄﻪ ﺗﺮﺗﻴﺐ ﺟﺰﻳﯽ‬
‫‪ ‬ﺭﺍ ﺭﻭﯼ ﻣﺠﻤﻮﻋﻪ }‪ X + = X È {^, ^¢‬ﺗﻌﺮﻳﻒ ﻣﯽﮐﻨﻴﻢ‪ .‬ﺩﺭ ﺭﺍﺑﻄﻪ ‪ v  w ، ‬ﺑﻪ ﺍﻳﻦ ﻣﻌﻨﯽ ﺍﺳﺖ ﮐﻪ ﺩﺭ ﮔﺮﺍﻑ ﺟﻬﺖ‬
‫ﺩﺍﺭ ﺍﺯ ‪ ‬ﻣﺴﻴﺮﯼ ﺑﻪ ‪ ‬ﻭﺟﻮﺩ ﺩﺍﺭﺩ‪ ^ .‬ﻋﻨﺼﺮ ﮐﻤﻴﻨﻪ ﻣﺠﻤﻮﻋﻪ ﺑﻮﺩﻩ ﻭ ﺑﻪ ﺍﺯﺍﯼ ﻫﺮ ﻋﻨﺼﺮ ﺩﺭ ‪ x Î X +‬ﺩﺍﺭﻳﻢ ‪. ^  x‬‬
‫‪ ^¢‬ﻋﻨﺼﺮ ﺑﻴﺸﻴﻨﻪ ﻣﺠﻤﻮﻋﻪ ﺑﻮﺩﻩ ﻭ ﺑﻪ ﺍﺯﺍﯼ ﻫﺮ ﻋﻨﺼﺮ ﺩﺭ ‪ x Î X +‬ﺩﺍﺭﻳﻢ ‪ . x  ^¢‬ﻳﮏ ﺑﺮﭼﺴﺐﮔﺬﺍﺭﯼ ﺳﺎﺯﮔﺎﺭ ﺩﺭ ‪X +‬‬
‫ﺍﻧﺘﺴﺎﺏ ﻣﻘﺎﺩﻳﺮ ]‪ g a Î [0,1‬ﺑﻪ ﻫﺮ ﻋﻨﺼﺮ ‪ X +‬ﺍﺳﺖ ﺑﻪ ﮔﻮﻧﻪﺍﯼ ﮐﻪ ‪ g ^¢ = 1 ، g ^ = 0‬و ‪ g a £ g b‬ﺍﮔﺮ ‪ a  b‬ﺩﺭ‬
‫ﺗﺮﺗﻴﺐ ﺟﺰﻳﯽ ﺑﺎﺷﺪ‪.‬‬
‫ﻟﻢ ‪ .١‬ﻓﺮﺽ ﮐﻨﻴﻢ ﮐﻪ ) ‪ G (V , E‬ﻳﮏ ﮔﺮﺍﻑ ﺩﻭﺑﺨﺸﯽ ﮐﺎﻣﻞ ﺑﺎ ﺑﺨﺶﻫﺎﯼ ‪ X‬ﻭ ‪ Y‬ﺑﺎﺷﺪ‪ .‬ﻫﻤﭽﻨﻴﻦ ‪ G‬ﺷﺎﻣﻞ ﺗﻄﺎﺑﻖ‬
‫ﮐﺎﻣﻞ ﻳﮑﺘﺎﯼ ‪ M‬ﺑﺎﺷﺪ‪ .‬ﺭﺍﺑﻄﻪ ﺗﺮﺗﻴﺐ ﺟﺰﻳﯽ ‪ ‬ﺭﺍ ﺩﺭ ‪ X +‬ﺩﺭ ﻧﻈﺮ ﻣﯽﮔﻴﺮﻳﻢ‪ .‬ﺧﺮﻭﺟﯽ ﭼﺎﻧﻪﺯﻧﯽ } ‪ {g v :v Î V‬ﻳﮏ ﺗﻌﺎﺩﻝ‬
‫ﭼﺎﻧﻪﺯﻧﯽ ﺩﺭ ﮔﺮﺍﻑ ﺍﺳﺖ ﺍﮔﺮ } ‪ {g v : v Î X +‬ﻳﮏ ﺑﺮﭼﺴﺐ ﮔﺬﺍﺭﯼ ﺳﺎﺯﮔﺎﺭ ﺑﺎﺷﺪ‪.‬‬
‫ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﻟﻢ ‪ ١‬ﻣﯽﺗﻮﺍﻥ ﺑﺎ ﻣﺤﺎﺳﺒﻪ ﻳﮏ ﺑﺮﭼﺴﺐ ﮔﺬﺍﺭﯼ ﺳﺎﺯﮔﺎﺭ ﻳﮏ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﺑﺮﺍﯼ ﮔﺮﺍﻑ ﻣﺤﺎﺳﺒﻪ ﻧﻤﻮﺩ‪ .‬ﻻﺯﻡ ﺑﻪ‬
‫ﺫﮐﺮ ﺍﺳﺖ ﮐﻪ ﻣﺤﺎﺳﺒﻪ ﺑﺮﭼﺴﺐﮔﺬﺍﺭﯼ ﺳﺎﺯﮔﺎﺭ ﺩﺭ ﺯﻣﺎﻥ ﭼﻨﺪﺟﻤﻠﻪﺍﯼ ﻭ ﺑﻪ ﺻﻮﺭﺕ ﺗﮑﺮﺍﺭﯼ ﺍﻓﺰﺍﻳﺸﯽ ﺑﺎ ﺷﺮﻭﻉ ﺍﺯ ‪ g ^ = 0‬و‬
‫‪ g ^¢ = 1‬ﻗﺎﺑﻞ ﺣﻞ ﺍﺳﺖ ]‪.[٤‬‬
‫‪ .٢ .٤‬ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﺩﺭ ﮔﺮﺍﻑ ﺩﻭﺑﺨﺸﯽ ﻋﻤﻮﻣﯽ‬
‫ﻗﻀﻴﻪ ‪ .١‬ﺑﺮﺍﯼ ﻫﺮ ﮔﺮﺍﻑ ﺩﻭﺑﺨﺸﯽ ‪ G‬ﻭ ﻫﺮ ﺗﻄﺎﺑﻖ ‪ M‬ﺩﺭ ﺁﻥ‪ ،‬ﻳﮏ ﺧﺮﻭﺟﯽ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﺍﮔﺮ ﻭ‬
‫ﻓﻘﻂ ﺍﮔﺮ ‪ M‬ﻳﮏ ﺗﻄﺎﺑﻖ ﺑﻴﺸﻴﻨﻪ ﺑﺎﺷﺪ‪.‬‬
‫ﺷﻬﻮﺩ ﺍﺛﺒﺎﺕ ﺍﻳﻦ ﻗﻀﻴﻪ ﺑﻪ ﺍﻳﻦ ﺻﻮﺭﺕ ﺍﺳﺖ‪ .‬ﻃﺒﻖ ﺍﻓﺮﺍﺯ ‪ [٦] Edmonds-Galli‬ﻣﯽﺗﻮﺍﻥ ﻫﺮ ﮔﺮﺍﻑ ﺭﺍ ﺑﻪ ‪ ٣‬ﺑﺨﺶ ﺍﻓﺮﺍﺯ‬
‫ﻧﻤﻮﺩ‪ .‬ﺑﺨﺶ ﺍﻭﻝ ‪ D‬ﺑﺨﺸﯽ ﺍﺳﺖ ﮐﻪ ﻫﺮ ﻧﻮﺩ ‪ v Î D‬ﺩﺭ ﻫﺮ ﺗﻄﺎﺑﻖ ﺑﻴﺸﻴﻨﻪ ﺍﺯ ﮔﺮﺍﻑ ﻭﺟﻮﺩ ﻧﺪﺍﺭﺩ‪ .‬ﺑﺨﺶ ﺩﻭﻡ ‪ A‬ﺷﺎﻣﻞ‬
‫ﻫﻤﻪ ﻧﻮﺩﻫﺎﻳﯽ ﺍﺳﺖ ﮐﻪ ﻳﮏ ﻫﻤﺴﺎﻳﻪ ﺩﺭ ﻣﺠﻤﻮﻋﻪ ‪ D‬ﺩﺍﺭﻧﺪ ﻭ ‪ C = V - A - D‬ﻣﺠﻤﻮﻋﻪﺍﯼ ﺍﺯ ﺭﺍﺱﻫﺎﺳﺖ ﮐﻪ ﻳﮏ ﺗﻄﺎﺑﻖ‬
‫ﮐﺎﻣﻞ ﺑﺮﺍﯼ ﺁﻧﻬﺎ ﻭﺟﻮﺩ ﺩﺍﺭﺩ‪ .‬ﺑﻪ ﻋﻨﻮﺍﻥ ﻣﺜﺎﻝ ﺩﺭ ﮔﺮﺍﻑ ﺷﮑﻞ ‪ A = {v} ، D = {u,w} ، ١‬ﻭ }‪ . C = {x, y‬ﭼﻮﻥ ﻣﺠﻤﻮﻋﻪ‬
‫‪ D‬ﺩﺭ ﮔﺮﺍﻑﻫﺎﯼ ﺩﻭ ﺑﺨﺸﯽ ﻳﮏ ﻣﺠﻤﻮﻋﻪ ﻣﺴﺘﻘﻞ ﺍﻳﺠﺎﺩ ﻣﯽﮐﻨﺪ ﺩﺭ ﺍﻳﻨﺼﻮﺭﺕ ﺑﻪ ﺍﺯﺍﯼ ﻫﺮ ‪ g v » 0 ، v Î D‬ﻭ ﺑﺮﺍﯼ ﻫﺮ‬
‫‪ g v » 1 ، v Î A‬ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪ .‬ﻣﺠﻤﻮﻋﻪ ‪ C‬ﭼﻮﻥ ﻳﮏ ﺗﻄﺎﺑﻖ ﮐﺎﻣﻞ ﺩﺍﺭﺩ ﻣﯽﺗﻮﺍﻥ ﻳﮏ ﺧﺮﻭﺟﯽ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﺑﺮ ﺍﺳﺎﺱ ﻟﻢ‬
‫‪ ١‬ﻣﺤﺎﺳﺒﻪ ﻧﻤﻮﺩ‪.‬‬
‫‪ .٥‬ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﺩﺭ ﺷﺒﮑﻪﻫﺎﯼ ﻋﻤﻮﻣﯽ‬
‫‪ .١ .٥‬ﺗﺎﺛﻴﺮ ﻧﻮﻉ ﺗﺎﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ﺩﺭ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ‬
‫ﺩﺭ ﺍﻳﻦ ﻗﺴﻤﺖ ﺑﺮ ﺍﺳﺎﺱ ﻧﻮﻉ ﺗﺎﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ﺑﻪ ﺑﺮﺭﺳﯽ ﺷﺮﺍﻳﻂ ﻭﺟﻮﺩ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﺩﺭ ﮔﺮﺍﻑﻫﺎﯼ ﻋﻤﻮﻣﯽ‬
‫ﻣﯽﭘﺮﺩﺍﺯﻳﻢ]‪. [٥‬‬
‫ﻗﻀﻴﻪ ‪ .١‬ﺍﮔﺮ ﺗﺎﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ﻫﻤﻪ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺧﻄﯽ ﺍﻓﺰﺍﻳﺸﯽ ﺑﺎﺷﺪ‪ ،‬ﺩﺭ ﺍﻳﻦ ﺻﻮﺭﺕ ﻳﮏ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﻳﮑﺘﺎ ﻭﺟﻮﺩ ﺩﺍﺭﺩ‬
‫ﮐﻪ ﺩﺭ ﺁﻥ ﺳﻮﺩ ﻫﺮ ﻳﺎﻝ ﺑﻪ ﻃﻮﺭ ﻣﺴﺎﻭﯼ ﺑﻴﻦ ﻫﺮ ﺩﻭ ﻧﻮﺩ ﻣﺠﺎﻭﺭ ﺁﻥ ﺗﻘﺴﻴﻢ ﻣﯽﺷﻮﺩ‪.‬‬
‫‪١‬‬
‫ﺍﺛﺒﺎﺕ‪ .‬ﻓﺮﺽ ﮐﻨﻴﻢ ﺗﺎﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ﺑﺮﺍﯼ ﺑﺎﺯﻳﮑﻦ ‪ u‬ﺑﺮﺍﺑﺮ ﺑﺎ ‪ U u ( x) = ax + b‬ﻭ ﺑﺎﺯﻳﮑﻦ ‪ U v ( x) = a¢x + b¢ v‬ﺑﺎﺷﺪ‪.‬‬
‫ﺑﺎ ﺟﺎﻳﮕﺬﺍﺭﯼ ﺭﻭﺍﺑﻂ ﺍﺯ ﺗﻌﺮﻳﻒ ‪ ٢‬ﻭ ﻣﺸﺘﻖ ﮔﻴﺮﯼ ﺩﺍﺭﻳﻢ‪:‬‬
‫))‪as (u ) = U u (a s (u , e) + x) - U u (a s (u, e‬‬
‫))‪bs (v) = U v (a s (v, e) + c( x) - x) -U v (a s (v, e‬‬
‫‪= a(a s (u, e) + x) + c - a (a s (u, e)) - c‬‬
‫‪= a¢(a s (v, e) + c( x) - x) + c¢ - a¢a s (v, e) - c¢‬‬
‫‪= ax‬‬
‫‪= a¢c( x) - a¢x‬‬
‫)‪f = as (u ) bs (v‬‬
‫‪2‬‬
‫‪= ax(a¢c( x) - a¢x) = aa¢c( x) x - aa¢x 2‬‬
‫‪f ¢ = aa¢c( x) - 2aa¢x‬‬
‫)‪f ¢ = 0 Þ aa¢c( x) - 2aa¢x = 0 Þ x = c( x‬‬
‫ﻗﻀﻴﻪ ﺯﻳﺮ ﺑﻪ ﻭﺟﻮﺩ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﺑﺎ ﺗﻮﺍﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ﻋﻤﻮﻣﯽ ﻣﯽﭘﺮﺩﺍﺯﺩ‪.‬‬
‫‪١‬‬
‫‪٢‬‬
‫ﺍﺛﺒﺎﺕ ﺍﻳﻦ ﻗﻀﻴﻪ ﺩﺭ ﻣﻘﺎﻟﻪ ﺍﺻﻠﯽ ﺁﻭﺭﺩﻩ ﻧﺸﺪﻩ ﺍﺳﺖ‪.‬‬
‫‪ ٢‬ﺩﺭ ﻣﻘﺎﻟﻪ ﺍﺻﻠﯽ ﺍﻳﻦ ﻭﻳﮋﮔﻴﻬﺎ ﺑﺮﺍﯼ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺴﺒﯽ ﺑﺮﺭﺳﯽ ﺷﺪﻩ ﺍﺳﺖ ﻭ ﻣﺎ ﺩﺭ ﺍﻳﻦ ﺟﺎ ﻫﻤﺎﻥ ﺭﻭﺵ ﺭﺍ ﺑﺮﺍﯼ ﺣﺎﻟﺖ ﻧﺶ ﺗﻌﻤﻴﻢ ﺩﺍﺩﻩﺍﻳﻢ‬
‫‪n‬‬
‫ﻗﻀﻴﻪ ‪ .٢‬ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﺩﺭ ﻫﺮ ﺷﺒﮑﻪ ﺍﺟﺘﻤﺎﻋﯽ ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﻫﺮﮔﺎﻩ ﺗﺎﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ﻫﻤﻪ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺍﻓﺰﺍﻳﺸﯽ‪ ،‬ﻣﺤﺪﺏ ﻭ‬
‫ﺩﻭ ﺑﺎﺭ ﻣﺸﺘﻖ ﭘﺬﻳﺮ ﺑﺎﺷﺪ‪.‬‬
‫ﺩﺭ ﻭﺍﻗﻊ ﺷﺮﻁ ﮐﺎﻓﯽ ﺍﻳﻦ ﺍﺳﺖ ﮐﻪ ﻫﻤﻪ ﺗﻮﺍﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺩﺭ ﺷﺮﻁ ﭘﻴﻮﺳﺘﮕﯽ ﺯﻳﺮ ﺻﺪﻕ ﮐﻨﻨﺪ‪:‬‬
‫ﺷﺮﻁ ‪ .١‬ﻓﺮﺽ ﮐﻨﻴﻢ ‪ s Î Âm‬ﺣﺎﻟﺖ ﻣﺪﻝ ﭼﺎﻧﻪﺯﻧﯽ ﺑﺎﺷﺪ ﻭ )‪ e = (u , v‬ﻳﺎﻟﯽ ﺩﺭ ﮔﺮﺍﻑ ﺑﺎﺷﺪ‪ .‬ﺑﻪ ﺍﺯﺍﯼ ﻫﺮ ‪ e > 0‬ﻭﺟﻮﺩ‬
‫ﺩﺍﺷﺘﻪ ﺑﺎﺷﺪ‬
‫‪ d > 0‬ﺑﻪ ﻃﻮﺭﻳﮑﻪ ﺑﺮﺍﯼ ﻫﺮ ﺣﺎﻟﺖ ﭼﺎﻧﻪﺯﻧﯽ‬
‫‪t‬‬
‫ﮐﻪ‬
‫‪ | a t (u, e) - a s (u, e) | < d‬ﻭ‬
‫‪ | a t (v, e) - a s (v, e) | < d‬ﺑﺮﻗﺮﺍﺭ ﺑﺎﺷﺪ‪ ،‬ﺩﺍﺷﺘﻪ ﺑﺎﺷﻴﻢ ‪. | yt (u, e) - ys (u , e) | < e‬‬
‫ﺭﻭﻧﺪ ﺍﺛﺒﺎﺕ ﺑﻪ ﺍﻳﻦ ﺻﻮﺭﺕ ﺍﺳﺖ ﮐﻪ ﺛﺎﺑﺖ ﻣﯽﮐﻨﻴﻢ ﺍﮔﺮ ﺷﺮﻁ ﭘﻴﻮﺳﺘﮕﯽ ‪ ١‬ﺻﺎﺩﻕ ﺑﺎﺷﺪ‪ ،‬ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﻭﺟﻮﺩ ﺧﻮﺍﻫﺪ‬
‫ﺩﺍﺷﺖ‪ .‬ﺳﭙﺲ ﺛﺎﺑﺖ ﻣﯽﮐﻨﻴﻢ ﮐﻪ ﻫﺮ ﺗﺎﺑﻊ ﺍﻓﺰﺍﻳﺸﯽ‪ ،‬ﻣﺤﺪﺏ ﻭ ﺩﻭ ﺑﺎﺭ ﻣﺸﺘﻖ ﭘﺬﻳﺮ ﮐﻪ ﭼﺎﻧﻪﺯﻧﯽ ﺑﺮ ﺍﺳﺎﺱ ﺭﺍﻩﺣﻞ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ‬
‫ﺻﻮﺭﺕ ﮔﻴﺮﺩ ﺩﺭ ﺷﺮﻁ ﭘﻴﻮﺳﺘﮕﯽ ‪ ١‬ﺻﺪﻕ ﻣﯽﮐﻨﺪ‪ .‬ﺑﺎ ﺍﺛﺒﺎﺕ ﺍﻳﻦ ﺩﻭ ﻟﻢ ﻗﻀﻴﻪ ‪ ٢‬ﺛﺎﺑﺖ ﻣﯽﺷﻮﺩ‪.‬‬
‫ﻟﻢ ‪ .١‬ﺍﮔﺮ ﺷﺮﻁ ﭘﻴﻮﺳﺘﮕﯽ ‪ ١‬ﺑﺮﺍﯼ ﺗﻮﺍﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ﺑﺎﺯﻳﮑﻨﺎﻧﯽ ﮐﻪ ﺩﺭ ﻳﮏ ﺷﺒﮑﻪ ﺍﺟﺘﻤﺎﻋﯽ ﺑﻪ ﺭﻭﺵ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﺑﻪ ﭼﺎﻧﻪﺯﻧﯽ‬
‫ﻣﯽﭘﺮﺩﺍﺯﻧﺪ ﺻﺎﺩﻕ ﺑﺎﺷﺪ ﺩﺭ ﺍﻳﻨﺼﻮﺭﺕ ﻳﮏ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﻭﺟﻮﺩ ﺧﻮﺍﻫﺪ ﺩﺍﺷﺖ‪.‬‬
‫ﺍﺛﺒﺎﺕ‪ .‬ﺗﺎﺑﻊ ‪ f : [0,1]m ® [0,1]m‬ﺭﺍ ﮐﻪ ﻫﺮ ﺣﺎﻟﺖ ﭼﺎﻧﻪﺯﻧﯽ ‪ s‬ﺭﺍ ﺣﺎﻟﺖ ﺩﻳﮕﺮ ) ‪ f (s‬ﻧﮕﺎﺷﺖ ﻣﯽﮐﻨﺪ ﺗﻌﺮﻳﻒ ﻣﯽﮐﻨﻴﻢ‪.‬‬
‫ﺑﺎ ﺩﺍﺷﺘﻦ )‪ ، e = (u , v‬ﻣﯽﺗﻮﺍﻧﻴﻢ ﺣﺎﻟﺖ ﻳﮑﺘﺎﯼ ‪ t‬ﺭﺍ ﺑﻪ ﮔﻮﻧﻪﺍﯼ ﺍﻳﺠﺎﺩ ﮐﻨﻴﻢ ﮐﻪ ﺩﺭ ﺁﻥ ﺍﺭﺯﺵ ﻫﺮ ﻳﺎﻝ )‪ e = (u, v‬ﺩﺭ ﺣﺎﻟﺖ‬
‫‪ t‬ﻣﻘﺪﺍﺭﯼ ﺍﺳﺖ ﮐﻪ ﺍﮔﺮ ﺩﻭ ﺑﺎﺯﻳﮑﻦ ‪ u‬ﻭ ‪ v‬ﺩﺭ ﺣﺎﻟﺖ ‪ s‬ﺑﺮ ﺭﻭﯼ ﻳﺎﻝ ﻧﺎﭘﺎﻳﺪﺍﺭ )‪ e = (u, v‬ﺩﻭﺑﺎﺭﻩ ﺑﻪ ﭼﺎﻧﻪﺯﻧﯽ ﺑﭙﺮﺩﺍﺯﻧﺪ ﺑﻪ‬
‫ﺩﺳﺖ ﻣﯽﺁﻭﺭﻧﺪ‪ .‬ﺑﻪ ﻋﺒﺎﺭﺕ ﺩﻳﮕﺮ ﺑﻪ )‪ f (s ) . xt (u , e) = ys (u , e‬ﺭﺍ ‪ t‬ﻗﺮﺍﺭ ﻣﯽﺩﻫﻴﻢ‪ .‬ﺑﺎ ﺍﻳﻦ ﺗﻌﺮﻳﻒ ﻣﺸﺨﺺ ﺍﺳﺖ ﮐﻪ‬
‫) ‪» f (s‬ﺑﻬﺘﺮﻳﻦ ﭘﺎﺳﺦ« ﺑﺮﺍﯼ ‪ s‬ﻣﯽﺑﺎﺷﺪ‪ .‬ﻣﺠﻤﻮﻋﻪ ‪ [0,1]m‬ﻳﮏ ﻣﺠﻤﻮﻋﻪ ﺑﺴﺘﻪ‪ ،‬ﻣﺤﺪﻭﺩ ﻭ ﻣﺤﺪﺏ ﺍﺳﺖ‪ .‬ﺍﮔﺮ ﺛﺎﺑﺖ ﮐﻨﻴﻢ‬
‫ﮐﻪ ‪ f‬ﻳﮏ ﺗﺎﺑﻊ ﭘﻴﻮﺳﺘﻪ ﺍﺳﺖ‪ ،‬ﻃﺒﻖ ﻗﻀﻴﻪ ﻧﻘﻄﻪ ﺛﺎﺑﺖ ‪ Brouwer‬ﺣﺎﻟﺖ ﭼﺎﻧﻪﺯﻧﯽ ‪ s‬ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﮐﻪ ‪ f ( s) = s‬ﻭ ﺍﻳﻦ‬
‫ﺣﺎﻟﺖ‪ ،‬ﺣﺎﻟﺖ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﺍﺳﺖ‪ .‬ﻟﺬﺍ ﻟﻢ ‪ ٢‬ﺍﺛﺒﺎﺕ ﻗﻀﻴﻪ ﺭﺍ ﮐﺎﻣﻞ ﻣﯽﮐﻨﺪ‪.‬‬
‫‪n‬‬
‫ﻟﻢ ‪ .٢‬ﺗﺎﺑﻊ ‪ f‬ﻣﻌﺮﻓﯽ ﺷﺪﻩ ﺩﺭ ﻟﻢ ‪ ١‬ﭘﻴﻮﺳﺘﻪ ﺍﺳﺖ ﺍﮔﺮ ﻭ ﻓﻘﻂ ﺍﮔﺮ ‪ f‬ﺩﺭ ﺷﺮﻁ ﭘﻴﻮﺳﺘﮕﯽ ‪ ١‬ﺻﺪﻕ ﮐﻨﺪ‪.‬‬
‫ﻟﻢ ‪ .٣‬ﺷﺮﻁ ﭘﻴﻮﺳﺘﮕﯽ ‪ ١‬ﺑﺮﺍﯼ ﻫﻤﻪ ﺗﻮﺍﺑﻊ ﺍﻓﺰﺍﻳﺸﯽ‪ ،‬ﻣﺤﺪﺏ ﻭ ﺩﻭ ﺑﺎﺭ ﻣﺸﺘﻖ ﭘﺬﻳﺮ ﻭﻗﺘﯽ ﭼﺎﻧﻪﺯﻧﯽ ﺍﺯ ﺭﺍﻩﺣﻞ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ‬
‫ﭘﻴﺮﻭﯼ ﻣﯽﮐﻨﻨﺪ ﺻﺎﺩﻕ ﺍﺳﺖ‪.‬‬
‫ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﻟﻢﻫﺎﯼ ‪ ١‬ﻭ ‪ ٣‬ﻗﻀﻴﻪ ‪ ٢‬ﺍﺛﺒﺎﺕ ﻣﯽﺷﻮﺩ‪.‬‬
‫‪ .٢ .٥‬ﺗﺎﺛﻴﺮﺳﺎﺧﺘﺎﺭ ﺷﺒﮑﻪ ﺩﺭ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ‬
‫ﻫﻤﺎﻧﻄﻮﺭ ﮐﻪ ﺩﺭ ﺷﺒﮑﻪ ﺷﮑﻞ ‪ ١‬ﺑﺮﺭﺳﯽ ﺷﺪ‪ ،‬ﺳﺎﺧﺘﺎﺭ ﺷﺒﮑﻪ ﻧﻘﺶ ﻣﻬﻤﯽ ﺩﺭ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﺩﺍﺭﺩ‪ .‬ﺩﺭ ﺍﻳﻦ ﺑﺨﺶ ﻧﺸﺎﻥ‬
‫ﺩﺍﺩﻩ ﻣﯽﺷﻮﺩ ﮐﻪ ﺑﺎ ﺩﺍﺷﺘﻦ ﻓﺮﺿﻴﺎﺗﯽ‪ ،‬ﺗﻮﭘﻮﻟﻮﮊﯼ ﺷﺒﮑﻪ ﺗﺎﺛﻴﺮﯼ ﺩﺭ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﻧﺪﺍﺭﺩ‪ .‬ﺩﺭ ﺍﺩﺍﻣﻪ ﻓﺮﺽ ﻣﯽﮐﻨﻴﻢ ﮐﻪ‬
‫ﻫﻤﻪ ﺭﺍﺱﻫﺎ ﺩﺍﺭﺍﯼ ﺗﺎﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ﻳﮑﺴﺎﻥ )‪ U (x‬ﺑﻮﺩﻩ ﻭ ﺍﺭﺯﺵ ﻫﺮ ﻳﺎﻝ ﺑﺮﺍﺑﺮ ﺑﺎ ﻳﮏ ﻣﯽﺑﺎﺷﺪ‪.‬‬
‫ﻗﻀﻴﻪ ‪ .٣‬ﻓﺮﺽ ﮐﻨﻴﻢ ﮐﻪ )‪ U (x‬ﺍﻓﺰﺍﻳﺸﯽ‪ ،‬ﻣﺤﺪﺏ ﻭ ﺩﻭ ﺑﺎﺭ ﻣﺸﺘﻖ ﭘﺬﻳﺮ ﺑﺎﺷﺪ‪ .‬ﻫﻤﭽﻨﻴﻦ ﺑﻪ ﺍﺯﺍﯼ ﻋﺪﺩ ﺛﺎﺑﺖ ‪ K‬ﺩﺍﺷﺘﻪ‬
‫)‪U ( x) - U (0‬‬
‫ﺑﺎﺷﻴﻢ ]‪< Kx, x Î [0,1‬‬
‫)‪U ¢( x‬‬
‫‪ .‬ﻫﻤﭽﻨﻴﻦ )‪ U ¢¢( x) £ e ( x)U ¢( x‬ﺑﺮﺍﯼ ﺗﺎﺑﻊ ﮐﺎﻫﺸﯽ )‪ e (x‬ﺑﺮﻗﺮﺍﺭ ﺑﺎﺷﺪ‪.‬‬
‫ﻓﺮﺽ ﮐﻨﻴﻢ ﮐﻪ ‪ s‬ﻳﮏ ﺣﺎﻟﺖ ﭼﺎﻧﻪﺯﻧﯽ ﺩﺭ ﺑﺎﺯﯼ ﺑﺎﺷﺪ ﻭ )‪ e = (u , v‬ﻳﺎﻟﯽ ﺑﺎﺷﺪ ﮐﻪ ﺩﺭ ﺁﻥ ﺩﺭﺟﻪ ﻧﻮﺩﻫﺎﯼ ‪ u‬ﻭ ‪ v‬ﺑﻪ ﺍﺯﺍﯼ‬
‫‪1‬‬
‫ﻋﺪﺩ ﻣﺜﺒﺖ ‪ d‬ﺍﺯ ‪ ( K + 1)d + 1‬ﺑﺰﺭﮔﺘﺮ ﺑﺎﺷﺪ‪ .‬ﺩﺭ ﺍﻳﻨﺼﻮﺭﺕ ) ‪. | xs (u, e) - |< e (d‬‬
‫‪2‬‬
‫‪-1‬‬
‫ﺑﺮﺍﯼ ﻣﺜﺎﻝ ﺗﺎﺑﻊ ‪ U ( x) = x p‬ﺑﺮﺍﯼ ‪ 0 < p < 1‬ﺩﺭ ﺷﺮﺍﻳﻂ ﻗﻀﻴﻪ ‪ ٣‬ﺻﺪﻕ ﻣﯽﮐﻨﺪ ﮐﻪ ﺩﺭ ﺁﻥ ‪ K = p‬ﻭ‬
‫‪ . e ( x) = (1 - p ) / x‬ﺗﺎﺑﻊ )‪ U ( x) = log(1 + x‬ﺑﺎ ‪ K = 2‬ﻭ ‪ e ( x) = 1 / 1 + x‬ﺩﺭ ﺷﺮﺍﻳﻂ ﻗﻀﻴﻪ ﺻﺪﻕ ﻣﯽﮐﻨﺪ‪.‬‬
‫ﻻﺯﻡ ﺑﻪ ﺩﮐﺮ ﺍﺳﺖ ﮐﻪ ﺷﺮﺍﻳﻂ ﻗﻀﻴﻪ ﻣﻄﺎﺑﻖ ﺑﺎ ﻗﻀﻴﻪ ‪ ٢‬ﺑﻮﺩﻩ ﻭ ﻭﺟﻮﺩ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﺑﺪﻳﻬﯽ ﺍﺳﺖ‪.‬‬
‫ﺩﻭ ﻟﻢ ﺯﻳﺮ ﻭﻳﮋﮔﯽﻫﺎﯼ ﻣﻬﻢ ﺩﻳﮕﺮﯼ ﺭﺍ ﻧﻴﺰ ﻧﺸﺎﻥ ﻣﯽﺩﻫﻨﺪ ﮐﻪ ﺩﺭ ﺍﺛﺒﺎﺕ ﻗﻀﻴﻪ ‪ ٣‬ﻣﻔﻴﺪ ﻫﺴﺘﻨﺪ‪.‬‬
‫ﻟﻢ ‪ .٤‬ﻓﺮﺽ ﮐﻨﻴﻢ ﮐﻪ )‪ U (x‬ﺍﻓﺰﺍﻳﺸﯽ‪ ،‬ﻣﺤﺪﺏ ﻭ ﺩﻭ ﺑﺎﺭ ﻣﺸﺘﻖ ﭘﺬﻳﺮ ﺑﺎﺷﺪ‪ .‬ﻫﻤﭽﻨﻴﻦ ﺑﻪ ﺍﺯﺍﯼ ﻋﺪﺩ ﺛﺎﺑﺖ ‪ K‬ﺩﺍﺷﺘﻪ ﺑﺎﺷﻴﻢ‬
‫)‪U ( x) - U (0‬‬
‫]‪< Kx, x Î [0,1‬‬
‫)‪U ¢( x‬‬
‫‪1‬‬
‫‪1‬‬
‫‪. xs (v , e ) ³‬‬
‫‪ xs (u , e) ³‬ﻭ‬
‫ﺧﻮﺍﻫﻴﻢ ﺩﺍﺷﺖ‪:‬‬
‫‪K +1‬‬
‫‪K +1‬‬
‫ﻟﻢ ‪ .٥‬ﻓﺮﺽ ﮐﻨﻴﻢ ﮐﻪ )‪ U (x‬ﺍﻓﺰﺍﻳﺸﯽ‪ ،‬ﻣﺤﺪﺏ ﻭ ﺩﻭ ﺑﺎﺭ ﻣﺸﺘﻖ ﭘﺬﻳﺮ ﺑﺎﺷﺪ‪ s .‬ﻳﮏ ﺣﺎﻟﺖ ﭼﺎﻧﻪﺯﻧﯽ ﺩﺭ ﺑﺎﺯﯼ ﺑﺎﺷﺪ ﻭ‬
‫)‪ e = (u , v‬ﻳﮏ ﻳﺎﻝ ﺩﺭ ﺷﺒﮑﻪ ﻭ ‪ e‬ﻋﺪﺩﯼ ﻣﺜﺒﺖ ﺑﺎﺷﺪ‪ .‬ﻫﻤﭽﻨﻴﻦ ))‪ U ¢¢(a s (u, e)) £ eU ¢(a s (u , e‬ﻭ‬
‫))‪ . U ¢¢(a s (v, e)) £ eU ¢(a s (v, e‬ﺩﺭ ﺍﻳﻨﺼﻮﺭﺕ ﺍﮔﺮ ‪ u‬ﺩﺭ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻣﻘﺪﺍﺭ ‪ x‬ﺩﺭﻳﺎﻓﺖ ﮐﻨﺪ ) ‪ v‬ﻣﻘﺪﺍﺭ ‪( 1 - x‬‬
‫‪ .‬ﺩﺭ ﺍﻳﻨﺼﻮﺭﺕ ﺩﺭ ﺣﺎﻟﺖ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ‪ ، s‬ﺑﻪ ﺍﺯﺍﯼ ﻫﺮ ﻳﺎﻝ )‪e = (u, v‬‬
‫‪1‬‬
‫ﺍﻧﮕﺎﻩ ‪|< e‬‬
‫‪2‬‬
‫‪|x-‬‬
‫‪ .٦‬ﻧﺘﻴﺠﻪﮔﻴﺮﯼ‬
‫ﺩﺭ ﺍﻳﻦ ﻣﻘﺎﻟﻪ ﺑﻪ ﺗﻌﺮﻳﻒ ﻣﺴﺌﻠﻪ ﭼﺎﻧﻪﺯﻧﯽ ﺩﻭ ﻃﺮﻓﻪ ﻭ ﺗﻌﻤﻴﻢ ﺁﻥ ﺑﻪ ﺷﺒﮑﻪﺍﯼ ﺍﺯ ﺍﻓﺮﺍﺩ ﭘﺮﺩﺍﺧﺘﻴﻢ‪ .‬ﺩﻭ ﺭﺍﻩﺣﻞ ﺍﺻﻠﯽ ﺍﺭﺍﺋﻪ‬
‫ﺷﺪﻩ ﺑﺮﺍﯼ ﭼﺎﻧﻪﺯﻧﯽ ﺩﻭ ﻃﺮﻓﻪ ﻳﻌﻨﯽ ﺭﺍﻩﺣﻞ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﻭ ﺭﺍﻩﺣﻞ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺴﺒﯽ ﻗﺎﺑﻞ ﺑﺮﺭﺳﯽ ﺩﺭ ﺷﺒﮑﻪﺍﯼ ﺍﺯ ﺑﺎﺯﻳﮑﻨﺎﻥ‬
‫ﻫﺴﺘﻨﺪ‪ .‬ﻧﮑﺘﻪﺍﯼ ﮐﻪ ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﺍﻳﻦ ﺍﺳﺖ ﮐﻪ ﺳﺎﺧﺘﺎﺭ ﺷﺒﮑﻪ ﻭ ﻧﻮﻉ ﺗﺎﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ﺑﺎﺯﻳﮑﻨﺎﻥ ﻧﻘﺶ ﺗﻌﻴﻴﻦ ﮐﻨﻨﺪﻩﺍﯼ ﺩﺭ ﻭﺟﻮﺩ‬
‫ﻭ ﻋﺪﻡ ﻭﺟﻮﺩ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﺩﺍﺭﻧﺪ‪ .‬ﻧﺘﺎﻳﺞ ﺍﺻﻠﯽ ﺑﺮﺭﺳﯽ ﺷﺪﻩ ﺭﺍ ﻣﯽﺗﻮﺍﻥ ﺩﺭ ﻣﻮﺍﺭﺩ ﺯﻳﺮ ﺧﻼﺻﻪ ﻧﻤﻮﺩ‪:‬‬
‫·‬
‫·‬
‫·‬
‫·‬
‫ﺑﺮﺍﯼ ﺗﻮﺍﺑﻊ ﺳﻮﻣﻨﺪﯼ ﺧﻄﯽ‪ ،‬ﻣﯽﺗﻮﺍﻥ ﺑﺎ ﻣﺤﺎﺳﺒﻪ ﻗﺪﺭﺕ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﺑﺮ ﺍﺳﺎﺱ ﺑﺎﺯﻳﮑﻨﺎﻥ ﻣﺠﺎﻭﺭ ﺁﻥ ﺩﺭ ﺷﺒﮑﻪ‬
‫ﺁﺟﺘﻤﺎﻋﯽ‪ ،‬ﺳﻬﻢ ﺑﺎﺯﻳﮑﻦ ﺭﺍ ﺍﺯ ﻫﻢ ﻣﻌﺎﻣﻠﻪ ﺩﺭ ﺷﺒﮑﻪ ﻣﺤﺎﺳﺒﻪ ﻧﻤﻮﺩ‪.‬‬
‫ﺑﺮﺍﯼ ﺗﻮﺍﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ﺧﻄﯽ ﻳﮑﺘﺎ‪ ،‬ﺑﺮﺍﯼ ﻫﺮ ﺷﺒﮑﻪ ﺍﺟﺘﻤﺎﻋﯽ ﻳﮑﺘﺎ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ) ﻧﺶ ﻭ ﻧﺴﺒﯽ( ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﻭ‬
‫ﺗﻮﭘﻮﻟﻮﮊﯼ ﺷﺒﮑﻪ ﻧﻘﺸﯽ ﺩﺭ ﺗﻌﺎﺩﻝ ﻧﻬﺎﻳﯽ ﻧﺪﺍﺭﺩ‪.‬‬
‫ﻫﺮﮔﺎﻩ ﺗﺎﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ﻫﻤﻪ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺍﻓﺰﺍﻳﺸﯽ‪ ،‬ﻣﺤﺪﺏ ﻭ ﺩﻭ ﺑﺎﺭ ﻣﺸﺘﻖ ﭘﺬﻳﺮ ﺑﺎﺷﺪ‪ ،‬ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﺩﺭ ﻫﺮ‬
‫ﺷﺒﮑﻪ ﺍﺟﺘﻤﺎﻋﯽ ﻭﺟﻮﺩ ﺩﺍﺭﺩ‬
‫ﺑﺎ ﻓﺮﺽ ﺗﺎﺑﻊ ﺳﻮﺩﻣﻨﺪﯼ ﺍﻓﺰﺍﻳﺸﯽ‪ ،‬ﻣﺤﺪﺏ ﻭ ﺩﻭ ﺑﺎﺭ ﻣﺸﺘﻖ ﭘﺬﻳﺮ ﻳﮑﺴﺎﻥ ﺑﺮﺍﯼ ﻫﻤﻪ ﺑﺎﺯﻳﮑﻨﺎﻥ‪ ،‬ﺗﻮﭘﻮﻟﻮﮊﯼ ﺷﺒﮑﻪ‬
‫ﺗﺎﺛﻴﺮﯼ ﺩﺭ ﺗﻌﺎﺩﻝ ﭼﺎﻧﻪﺯﻧﯽ ﻧﺶ ﻧﺪﺍﺭﺩ‪.‬‬
‫ ﻣﺮﺍﺟﻊ‬.٧
[١] John Forbes Nash. The bargaining problem. Econometrica, ١٨(٢):١٥٥-١٦٢, ١٩٥٠
[٢] M.J. Osborne. “An introduction to game theory”, Oxford University Press New York, ٢٠٠٤
(Chapter ١٥)
[٣] N. Braun, T. Gautschi, A nash bargaining model for simple exchange networks. Social
Networks ٢٨(١) (٢٠٠٦) ١–٢٣
[٤] J. Kleinberg, E. Tardos, Balanced outcomes in social exchange networks. STOC ٢٠٠٨
[٥] T. Chakraborty , M. Kearns , Bargaining Solutions in a Social Network. T. Chakraborty. WINE
٢٠٠٨.
[٦] L. Lov´asz and M. Plummer. Matching Theory. North-Holland, ١٩٨٦.
١
Bargaining Problem
Nash Bargaining Solution
٣
Independence from Irrelevant Alternatives
٢