Stability of Equilibrial States and Limit Cycles in Sparsely Connected, Structurally Complex Boolean Nets

Stab ility of E q u ilibrial S t a t e s a n d Lim it Cycles in Sparsely Co n nected, Structurally Com p lex B o olean Nets C rayt on C. Walker Abst ract. Th roug h the ir na tural dyn ami cs , net work s of Boo lean fun ct ions ca n recompu te , or ret urn to , equilibrial states and lim it cycles from which t hey have been d isp laced. How often such return occurs measures one kind of be havi o ral stability. For a clas s of fu nc-t ionally homogeneous , spa rsely conn ected, disor derly st ruct ure d networks , t his paper examines how stability of equi libria l states a nd limi t cycles is affected by th e size of th e net , size of t he displ acem ent, an d t he functi on used in th e net. All functions in t he class ar e exam ined , a nd displ acement is vari ed over its full range. On t he whole, for an y given fun cti on , the effect of d isplaceme nt size ap pear s t o be qui te regular . However , from function to function , d isplacemen t effects vary widely. T hough t here ar e exceptions , at a given rela tive disp lacement , mos t fun ctions show de creas ing cycli c stability as nets beco me la rger. On t he other hand , t he data suggest that for many fun ct ion s , lar ger nets are more stable under small ab solute d isp lacem ents. 1. I ntroducti on Many-element mod els have long been used to und erstand complex real world systems. More recently, increased attention has been given to the st udy of many-element models in which element behavior can decisively influence overall syst em behavior (as an earl y example, see [1]). T he present paper continues this very-compl ex-system th eme using a class of dynamic Boolean nets to provide the objects of inte rest. An impo rtant characteristic of any dynamic system is t he extent to which its dynamics resist pertubations: its "behavioral stability." While behavioral stability is equ ally important in network systems, t …