Preferential Attachment Random Graphs with General Weight Function

Abstract

Start with graph _G_₀ ≡ {_V_₁, _V_₂} with one edge connecting the two vertices _V_₁, _V_₂. Now create a new vertex _V_₃ and attach it (i.e., add an edge) to _V_₁ or _V_₂ with equal probability. Set _G_₁ ≡ {_V_₁, _V_₂, _V_₃}. Let _G__n ≡ {_V_₁, _V_₂, . . . , _V__n+2} be the graph after _n_ steps, _n_ ≥ 0. For each _i_, 1 ≤ _i_ ≤ _n_+2, let _d__<em>n</em>(_i_) be the number of vertices in _G__<em>n</em> to which _V__<em>i</em> is connected. Now create a new vertex _V__<em>n</em>+3 and attach it to _V__<em>i</em> in _G__<em>n</em> with probability proportional to _w_(_d__<em>n</em>(_i_)), 1 ≤ _i_ ≤ _n_+2, where _w_(·) is a function from _N_ ≡ {1, 2, 3, . . .} to (0,∞). In this paper, some results on behavior of the degree sequence {_d__<em>n</em>(_i_)}_{<em>n</em>≥1,<em>i</em>≥1} and the empirical distribution <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uinm_a_10129150_o_ilf0001.gif"></inline-graphic> are derived. Our results indicate that the much discussed power-law growth of _d__<em>n</em>(_i_) and power law decay of <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uinm_a_10129150_o_ilf0002.gif"></inline-graphic> hold essentially only when the weight function _w_(·) is asymptotically linear. For example, if _w_(_x_) = _cx_² then for _i_ ≥ 1, lim_<em>n</em>_d__<em>n</em>(_i_) exists and is finite with probability (w.p.) 1 and <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uinm_a_10129150_o_ilf0003.gif"></inline-graphic>, and if _w_(_x_) = _cx_^p, 1/2 < _p_ < 1 then for _i_ ≥ 1, _d__<em>n</em>(_i_) grows like (log _n_)^q where _q_ = (1 − _p_)⁻¹. The main tool used in this paper is an embedding in continuous time of pure birth Markov chains.